P. 21 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 2001-2100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ax9v1 2001*	First of two weakened versions of ax9v 2000, with an extra dv condition on x , z, see comments there. (Contributed by BJ, 7-Dec-2020.)
|- ( x = y -> ( z e. x -> z e. y ) )
Theorem	ax9v2 2002*	Second of two weakened versions of ax9v 2000, with an extra dv condition on y , z see comments there. (Contributed by BJ, 7-Dec-2020.)
|- ( x = y -> ( z e. x -> z e. y ) )
Theorem	ax9 2003	Proof of ax-9 1999 from ax9v1 2001 and ax9v2 2002, proving sufficiency of the conjunction of the latter two weakened versions of ax9v 2000, which is itself a weakened version of ax-9 1999. (Contributed by BJ, 7-Dec-2020.) (Proof shortened by Wolf Lammen, 11-Apr-2021.)
|- ( x = y -> ( z e. x -> z e. y ) )
Theorem	elequ2 2004	An identity law for the non-logical predicate. (Contributed by NM, 21-Jun-1993.)
|- ( x = y -> ( z e. x <-> z e. y ) )
1.4.12  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13 The original axiom schemes of Tarski's predicate calculus are ax-4 1737, ax-5 1839, ax6v 1889, ax-7 1935, ax-8 1992, and ax-9 1999, together with rule ax-gen 1722. See mmset.html#compare 1722. They are given as axiom schemes B4 through B8 in [KalishMontague] p. 81. These are shown to be logically complete by Theorem 1 of [KalishMontague] p. 85. The axiom system of set.mm includes the auxiliary axiom schemes ax-10 2019, ax-11 2034, ax-12 2047, and ax-13 2246, which are not part of Tarski's axiom schemes. Each object-language instance of them is provable from Tarski's axioms, so they are logically redundant. However, they are conjectured not to be provable directly as schemes from Tarski's axiom schemes using only Metamath's direct substitution rule. They are used to make our system "metalogically complete" i.e. able to prove directly all possible schemes with wff and setvar variables, bundled or not, whose object-language instances are valid. (ax-12 2047 has been proved to be required; see http://us.metamath.org/award2003.html#9a. Metalogical independence of the other three are open problems.) (There are additional predicate calculus axiom schemes included in set.mm such as ax-c5 34168, but they can all be proved as theorems from the above.) Terminology: Two setvar (individual) metavariables are "bundled" in an axiom or theorem scheme when there is no distinct variable constraint ($d) imposed on them. (The term "bundled" is due to Raph Levien.) For example, the x and y in ax-6 1888 are bundled, but they are not in ax6v 1889. We also say that a scheme is bundled when it has at least one pair of bundled setvar variables. If distinct variable conditions are added to all setvar variable pairs in a bundled scheme, we call that the "principal" instance of the bundled scheme. For example, ax6v 1889 is the principal instance of ax-6 1888. Whenever a common variable is substituted for two or more bundled variables in an axiom or theorem scheme, we call the substitution instance "degenerate". For example, the instance -. A. x -. x = x of ax-6 1888 is degenerate. An advantage of bundling is ease of use since there are fewer distinct variable restrictions ($d) to be concerned with, and theorems are more general. There may be some economy in being able to prove facts about principal and degenerate instances simultaneously. A disadvantage is that bundling may present difficulties in translations to other proof languages, which typically lack the concept (in part because their variables often represent the variables of the object language rather than metavariables ranging over them). Because Tarski's axiom schemes are logically complete, they can be used to prove any object-language instance of ax-10 2019, ax-11 2034, ax-12 2047, and ax-13 2246. "Translating" this to Metamath, it means that Tarski's axioms can prove any substitution instance of ax-10 2019, ax-11 2034, ax-12 2047, or ax-13 2246 in which (1) there are no wff metavariables and (2) all setvar variables are mutually distinct i.e. are not bundled. In effect this is mimicking the object language by pretending that each setvar variable is an object-language variable. (There may also be specific instances with wff metavariables and/or bundling that are directly provable from Tarski's axiom schemes, but it isn't guaranteed. Whether all of them are possible is part of the still open metalogical independence problem for our additional axiom schemes.) It can be useful to see how this can be done, both to show that our additional schemes are valid metatheorems of Tarski's system and to be able to translate object-language instances of our proofs into proofs that would work with a system using only Tarski's original schemes. In addition, it may (or may not) provide insight into the conjectured metalogical independence of our additional schemes. The theorem schemes ax10w 2006, ax11w 2007, ax12w 2010, and ax13w 2013 are derived using only Tarski's axiom schemes, showing that Tarski's schemes can be used to derive all substitution instances of ax-10 2019, ax-11 2034, ax-12 2047, and ax-13 2246 meeting Conditions (1) and (2). (The "w" suffix stands for "weak version".) Each hypothesis of ax10w 2006, ax11w 2007, and ax12w 2010 is of the form ( x = y -> ( ph <-> ps ) ) where ps is an auxiliary or "dummy" wff metavariable in which x doesn't occur. We can show by induction on formula length that the hypotheses can be eliminated in all cases meeting Conditions (1) and (2). The example ax12wdemo 2012 illustrates the techniques (equality theorems and bound variable renaming) used to achieve this. We also show the degenerate instances for axioms with bundled variables in ax11dgen 2008, ax12dgen 2011, ax13dgen1 2014, ax13dgen2 2015, ax13dgen3 2016, and ax13dgen4 2017. (Their proofs are trivial, but we include them to be thorough.) Combining the principal and degenerate cases outside of Metamath, we show that the bundled schemes ax-10 2019, ax-11 2034, ax-12 2047, and ax-13 2246 are schemes of Tarski's system, meaning that all object-language instances they generate are theorems of Tarski's system. It is interesting that Tarski used the bundled scheme ax-6 1888 in an older system, so it seems the main purpose of his later ax6v 1889 was just to show that the weaker unbundled form is sufficient rather than an aesthetic objection to bundled free and bound variables. Since we adopt the bundled ax-6 1888 as our official axiom, we show that the degenerate instance holds in ax6dgen 2005. (Recall that in set.mm, the only statement referencing ax-6 1888 is ax6v 1889.) The case of sp 2053 is curious: originally an axiom scheme of Tarski's system, it was proved logically redundant by Lemma 9 of [KalishMontague] p. 86. However, the proof is by induction on formula length, and the scheme form A. x ph -> ph apparently cannot be proved directly from Tarski's other axiom schemes. The best we can do seems to be spw 1967, again requiring substitution instances of ph that meet Conditions (1) and (2) above. Note that our direct proof sp 2053 requires ax-12 2047, which is not part of Tarski's system.
Theorem	ax6dgen 2005	Tarski's system uses the weaker ax6v 1889 instead of the bundled ax-6 1888, so here we show that the degenerate case of ax-6 1888 can be derived. Even though ax-6 1888 is in the list of axioms used, recall that in set.mm, the only statement referencing ax-6 1888 is ax6v 1889. We later rederive from ax6v 1889 the bundled form as ax6 2251 with the help of the auxiliary axiom schemes. (Contributed by NM, 23-Apr-2017.)
|- -. A. x -. x = x
Theorem	ax10w 2006*	Weak version of ax-10 2019 from which we can prove any ax-10 2019 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. It is an alias of hbn1w 1973 introduced for labeling consistency. (Contributed by NM, 9-Apr-2017.) Use hbn1w 1973 instead. (New usage is discouraged.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( -. A. x ph -> A. x -. A. x ph )
Theorem	ax11w 2007*	Weak version of ax-11 2034 from which we can prove any ax-11 2034 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. Unlike ax-11 2034, this theorem requires that x and y be distinct i.e. are not bundled. It is an alias of alcomiw 1971 introduced for labeling consistency. (Contributed by NM, 10-Apr-2017.) Use alcomiw 1971 instead. (New usage is discouraged.)
|- ( y = z -> ( ph <-> ps ) )   =>    |- ( A. x A. y ph -> A. y A. x ph )
Theorem	ax11dgen 2008	Degenerate instance of ax-11 2034 where bundled variables x and y have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.)
|- ( A. x A. x ph -> A. x A. x ph )
Theorem	ax12wlem 2009*	Lemma for weak version of ax-12 2047. Uses only Tarski's FOL axiom schemes. In some cases, this lemma may lead to shorter proofs than ax12w 2010. (Contributed by NM, 10-Apr-2017.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) )
Theorem	ax12w 2010*	Weak version of ax-12 2047 from which we can prove any ax-12 2047 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. An instance of the first hypothesis will normally require that x and y be distinct (unless x does not occur in ph). For an example of how the hypotheses can be eliminated when we substitute an expression without wff variables for ph, see ax12wdemo 2012. (Contributed by NM, 10-Apr-2017.)
|- ( x = y -> ( ph <-> ps ) )   &    |- ( y = z -> ( ph <-> ch ) )   =>    |- ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) )
Theorem	ax12dgen 2011	Degenerate instance of ax-12 2047 where bundled variables x and y have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.)
|- ( x = x -> ( A. x ph -> A. x ( x = x -> ph ) ) )
Theorem	ax12wdemo 2012*	Example of an application of ax12w 2010 that results in an instance of ax-12 2047 for a contrived formula with mixed free and bound variables, ( x e. y /\ A. x z e. x /\ A. y A. z y e. x ), in place of ph. The proof illustrates bound variable renaming with cbvalvw 1969 to obtain fresh variables to avoid distinct variable clashes. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 14-Apr-2017.)
|- ( x = y -> ( A. y ( x e. y /\ A. x z e. x /\ A. y A. z y e. x ) -> A. x ( x = y -> ( x e. y /\ A. x z e. x /\ A. y A. z y e. x ) ) ) )
Theorem	ax13w 2013*	Weak version (principal instance) of ax-13 2246. (Because y and z don't need to be distinct, this actually bundles the principal instance and the degenerate instance ( -. x = y -> ( y = y -> A. x y = y ) ).) Uses only Tarski's FOL axiom schemes. The proof is trivial but is included to complete the set ax10w 2006, ax11w 2007, and ax12w 2010. (Contributed by NM, 10-Apr-2017.)
|- ( -. x = y -> ( y = z -> A. x y = z ) )
Theorem	ax13dgen1 2014	Degenerate instance of ax-13 2246 where bundled variables x and y have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.)
|- ( -. x = x -> ( x = z -> A. x x = z ) )
Theorem	ax13dgen2 2015	Degenerate instance of ax-13 2246 where bundled variables x and z have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.)
|- ( -. x = y -> ( y = x -> A. x y = x ) )
Theorem	ax13dgen3 2016	Degenerate instance of ax-13 2246 where bundled variables y and z have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.)
|- ( -. x = y -> ( y = y -> A. x y = y ) )
Theorem	ax13dgen4 2017	Degenerate instance of ax-13 2246 where bundled variables x, y, and z have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.) Reduce axiom usage. (Revised by Wolf Lammen, 10-Oct-2021.)
|- ( -. x = x -> ( x = x -> A. x x = x ) )
Theorem	ax13dgen4OLD 2018	Obsolete proof of ax13dgen4 2017 as of 10-Oct-2021. (Contributed by NM, 13-Apr-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( -. x = x -> ( x = x -> A. x x = x ) )
1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes) In this section we introduce four additional schemes ax-10 2019, ax-11 2034, ax-12 2047, and ax-13 2246 that are not part of Tarski's system but can be proved (outside of Metamath) as theorem schemes of Tarski's system. These are needed to give our system the property of "scheme completeness," which means that we can prove (with Metamath) all possible theorem schemes expressible in our language of wff metavariables ranging over object-language wffs, and setvar variables ranging over object-language individual variables. To show that these schemes are valid metatheorems of Tarski's system S2, above we proved from Tarski's system theorems ax10w 2006, ax11w 2007, ax12w 2010, and ax13w 2013, which show that any object-language instance of these schemes (emulated by having no wff metavariables and requiring all setvar variables to be mutually distinct) can be proved using only the schemes in Tarski's system S2. An open problem is to show that these four additional schemes are mutually metalogically independent and metalogically independent from Tarski's. So far, independence of ax-12 2047 from all others has been shown, and independence of Tarski's ax-6 1888 from all others has been shown; see items 9a and 11 on http://us.metamath.org/award2003.html.
1.5.1  Axiom scheme ax-10 (Quantified Negation)
Axiom	ax-10 2019	Axiom of Quantified Negation. Axiom C5-2 of [Monk2] p. 113. This axiom scheme is logically redundant (see ax10w 2006) but is used as an auxiliary axiom scheme to achieve scheme completeness. It means that x is not free in -. A. x ph. (Contributed by NM, 21-May-2008.) Use its alias hbn1 2020 instead if you must use it. Any theorem in first order logic (FOL) that contains only set variables that are all mutually distinct, and has no wff variables, can be proved *without* using ax-10 2019 through ax-13 2246, by invoking ax10w 2006 through ax13w 2013. We encourage proving theorems *without* ax-10 2019 through ax-13 2246 and moving them up to the ax-4 1737 through ax-9 1999 section. (New usage is discouraged.)
|- ( -. A. x ph -> A. x -. A. x ph )
Theorem	hbn1 2020	Alias for ax-10 2019 to be used instead of it. (Contributed by NM, 24-Jan-1993.) (Proof shortened by Wolf Lammen, 18-Aug-2014.)
|- ( -. A. x ph -> A. x -. A. x ph )
Theorem	hbe1 2021	The setvar x is not free in E. x ph. (Contributed by NM, 24-Jan-1993.)
|- ( E. x ph -> A. x E. x ph )
Theorem	hbe1a 2022	Dual statement of hbe1 2021. Modified version of axc7e 2133 with a universally quantified consequent. (Contributed by Wolf Lammen, 15-Sep-2021.)
|- ( E. x A. x ph -> A. x ph )
Theorem	nf5-1 2023	One direction of nf5 2116 can be proved with a smaller footprint on axiom usage. (Contributed by Wolf Lammen, 16-Sep-2021.)
|- ( A. x ( ph -> A. x ph ) -> F/ x ph )
Theorem	nf5i 2024	Deduce that x is not free in ph from the definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- ( ph -> A. x ph )   =>    |- F/ x ph
Theorem	nf5dv 2025*	Apply the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1710 changed. (Revised by Wolf Lammen, 18-Sep-2021.)
|- ( ph -> ( ps -> A. x ps ) )   =>    |- ( ph -> F/ x ps )
Theorem	nf5dh 2026	Deduce that x is not free in ps in a context. (Contributed by Mario Carneiro, 24-Sep-2016.) df-nf 1710 changed. (Revised by Wolf Lammen, 11-Oct-2021.)
|- ( ph -> A. x ph )   &    |- ( ph -> ( ps -> A. x ps ) )   =>    |- ( ph -> F/ x ps )
Theorem	nfe1 2027	The setvar x is not free in E. x ph. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/ x E. x ph
Theorem	nfa1 2028	The setvar x is not free in A. x ph. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1710 changed. (Revised by Wolf Lammen, 11-Sep-2021.) Remove dependency on ax-12 2047. (Revised by Wolf Lammen, 12-Oct-2021.)
|- F/ x A. x ph
Theorem	nfna1 2029	A convenience theorem particularly designed to remove dependencies on ax-11 2034 in conjunction with distinctors. (Contributed by Wolf Lammen, 2-Sep-2018.)
|- F/ x -. A. x ph
Theorem	nfia1 2030	Lemma 23 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.)
|- F/ x ( A. x ph -> A. x ps )
Theorem	nfnf1 2031	The setvar x is not free in F/ x ph. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-12 2047. (Revised by Wolf Lammen, 12-Oct-2021.)
|- F/ x F/ x ph
Theorem	modal-5 2032	The analogue in our predicate calculus of axiom (5) of modal logic S5. (Contributed by NM, 5-Oct-2005.)
|- ( -. A. x -. ph -> A. x -. A. x -. ph )
Theorem	nfe1OLD 2033	Obsolete proof of nfe1 2027 as of 6-Oct-2021. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
|- F/ x E. x ph
1.5.2  Axiom scheme ax-11 (Quantifier Commutation)
Axiom	ax-11 2034	Axiom of Quantifier Commutation. This axiom says universal quantifiers can be swapped. Axiom scheme C6' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Lemma 12 of [Monk2] p. 109 and Axiom C5-3 of [Monk2] p. 113. This axiom scheme is logically redundant (see ax11w 2007) but is used as an auxiliary axiom scheme to achieve metalogical completeness. (Contributed by NM, 12-Mar-1993.)
|- ( A. x A. y ph -> A. y A. x ph )
Theorem	alcoms 2035	Swap quantifiers in an antecedent. (Contributed by NM, 11-May-1993.)
|- ( A. x A. y ph -> ps )   =>    |- ( A. y A. x ph -> ps )
Theorem	hbal 2036	If x is not free in ph, it is not free in A. y ph. (Contributed by NM, 12-Mar-1993.)
|- ( ph -> A. x ph )   =>    |- ( A. y ph -> A. x A. y ph )
Theorem	alcom 2037	Theorem 19.5 of [Margaris] p. 89. (Contributed by NM, 30-Jun-1993.)
|- ( A. x A. y ph <-> A. y A. x ph )
Theorem	alrot3 2038	Theorem *11.21 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
|- ( A. x A. y A. z ph <-> A. y A. z A. x ph )
Theorem	alrot4 2039	Rotate four universal quantifiers twice. (Contributed by NM, 2-Feb-2005.) (Proof shortened by Fan Zheng, 6-Jun-2016.)
|- ( A. x A. y A. z A. w ph <-> A. z A. w A. x A. y ph )
Theorem	nfa2 2040	Lemma 24 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.) Remove dependency on ax-12 2047. (Revised by Wolf Lammen, 18-Oct-2021.)
|- F/ x A. y A. x ph
Theorem	hbald 2041	Deduction form of bound-variable hypothesis builder hbal 2036. (Contributed by NM, 2-Jan-2002.)
|- ( ph -> A. y ph )   &    |- ( ph -> ( ps -> A. x ps ) )   =>    |- ( ph -> ( A. y ps -> A. x A. y ps ) )
Theorem	excom 2042	Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) Remove dependencies on ax-5 1839, ax-6 1888, ax-7 1935, ax-10 2019, ax-12 2047. (Revised by Wolf Lammen, 8-Jan-2018.) (Proof shortened by Wolf Lammen, 22-Aug-2020.)
|- ( E. x E. y ph <-> E. y E. x ph )
Theorem	excomim 2043	One direction of Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) Remove dependencies on ax-5 1839, ax-6 1888, ax-7 1935, ax-10 2019, ax-12 2047. (Revised by Wolf Lammen, 8-Jan-2018.)
|- ( E. x E. y ph -> E. y E. x ph )
Theorem	excom13 2044	Swap 1st and 3rd existential quantifiers. (Contributed by NM, 9-Mar-1995.)
|- ( E. x E. y E. z ph <-> E. z E. y E. x ph )
Theorem	exrot3 2045	Rotate existential quantifiers. (Contributed by NM, 17-Mar-1995.)
|- ( E. x E. y E. z ph <-> E. y E. z E. x ph )
Theorem	exrot4 2046	Rotate existential quantifiers twice. (Contributed by NM, 9-Mar-1995.)
|- ( E. x E. y E. z E. w ph <-> E. z E. w E. x E. y ph )
1.5.3  Axiom scheme ax-12 (Substitution)
Axiom	ax-12 2047	Axiom of Substitution. One of the 5 equality axioms of predicate calculus. The final consequent A. x ( x = y -> ph ) is a way of expressing " y substituted for x in wff ph " (cf. sb6 2429). It is based on Lemma 16 of [Tarski] p. 70 and Axiom C8 of [Monk2] p. 105, from which it can be proved by cases. The original version of this axiom was ax-c15 34174 and was replaced with this shorter ax-12 2047 in Jan. 2007. The old axiom is proved from this one as theorem axc15 2303. Conversely, this axiom is proved from ax-c15 34174 as theorem ax12 2304. Juha Arpiainen proved the metalogical independence of this axiom (in the form of the older axiom ax-c15 34174) from the others on 19-Jan-2006. See item 9a at http://us.metamath.org/award2003.html. See ax12v 2048 and ax12v2 2049 for other equivalents of this axiom that (unlike this axiom) have distinct variable restrictions. This axiom scheme is logically redundant (see ax12w 2010) but is used as an auxiliary axiom scheme to achieve scheme completeness. (Contributed by NM, 22-Jan-2007.) (New usage is discouraged.)
|- ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) )
Theorem	ax12v 2048*	This is essentially axiom ax-12 2047 weakened by additional restrictions on variables. Besides axc11r 2187, this theorem should be the only one referencing ax-12 2047 directly. Both restrictions on variables have their own value. If for a moment we assume y could be set to x, then, after elimination of the tautology x = x, immediately we have ph -> A. x ph for all ph and x, that is ax-5 1839, a degenerate result. The second restriction is not necessary, but a simplification that makes the following interpretation easier to see. Since ph textually at most depends on x, we can look at it at some given 'fixed' y. This theorem now states that the truth value of ph will stay constant, as long as we 'vary x around y' only such that x = y still holds. Or in other words, equality is the finest grained logical expression. If you cannot differ two sets by =, you won't find a whatever sophisticated expression that does. One might wonder how the described variation of x is possible at all. Note that Metamath is a text processor that easily sees a difference between text chunks { x | -. x = x } and { y | -. y = y }. Our usual interpretation is to abstract from textual variations of the same set, but we are free to interpret Metamath's formalism differently, and in fact let x run through all textual representations of sets. Had we allowed ph to depend also on y, this idea is both harder to see, and it is less clear that this extra freedom introduces effects not covered by other axioms. (Contributed by Wolf Lammen, 8-Aug-2020.)
|- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) )
Theorem	ax12v2 2049*	It is possible to remove any restriction on ph in ax12v 2048. Same as Axiom C8 of [Monk2] p. 105. Use ax12v 2048 instead when sufficient. (Contributed by NM, 5-Aug-1993.) Removed dependencies on ax-10 2019 and ax-13 2246. (Revised by Jim Kingdon, 15-Dec-2017.) (Proof shortened by Wolf Lammen, 8-Dec-2019.)
|- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) )
Theorem	ax12vOLD 2050*	Obsolete proof of ax12v2 2049 as of 24-Mar-2021. (Contributed by NM, 5-Aug-1993.) Removed dependencies on ax-10 2019 and ax-13 2246. (Revised by Jim Kingdon, 15-Dec-2017.) (Proof shortened by Wolf Lammen, 8-Dec-2019.) (Proof shortened by Wolf Lammen, 7-Mar-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
|- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) )
Theorem	ax12vOLDOLD 2051*	Obsolete proof of ax12v 2048 as of 7-Mar-2021. (Contributed by NM, 5-Aug-1993.) Removed dependencies on ax-10 2019 and ax-13 2246. (Revised by Jim Kingdon, 15-Dec-2017.) (Proof shortened by Wolf Lammen, 8-Dec-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
|- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) )
Theorem	19.8a 2052	If a wff is true, it is true for at least one instance. Special case of Theorem 19.8 of [Margaris] p. 89. See 19.8v 1895 for a version with a dv condition requiring fewer axioms. (Contributed by NM, 9-Jan-1993.) Allow a shortening of sp 2053. (Revised by Wolf Lammen, 13-Jan-2018.) (Proof shortened by Wolf Lammen, 8-Dec-2019.)
|- ( ph -> E. x ph )
Theorem	sp 2053	Specialization. A universally quantified wff implies the wff without a quantifier Axiom scheme B5 of [Tarski] p. 67 (under his system S2, defined in the last paragraph on p. 77). Also appears as Axiom scheme C5' in [Megill] p. 448 (p. 16 of the preprint). This corresponds to the axiom (T) of modal logic. For the axiom of specialization presented in many logic textbooks, see theorem stdpc4 2353. This theorem shows that our obsolete axiom ax-c5 34168 can be derived from the others. The proof uses ideas from the proof of Lemma 21 of [Monk2] p. 114. It appears that this scheme cannot be derived directly from Tarski's axioms without auxiliary axiom scheme ax-12 2047. It is thought the best we can do using only Tarski's axioms is spw 1967. (Contributed by NM, 21-May-2008.) (Proof shortened by Scott Fenton, 24-Jan-2011.) (Proof shortened by Wolf Lammen, 13-Jan-2018.)
|- ( A. x ph -> ph )
Theorem	spi 2054	Inference rule reversing generalization. (Contributed by NM, 5-Aug-1993.)
|- A. x ph   =>    |- ph
Theorem	sps 2055	Generalization of antecedent. (Contributed by NM, 5-Jan-1993.)
|- ( ph -> ps )   =>    |- ( A. x ph -> ps )
Theorem	2sp 2056	A double specialization (see sp 2053). Another double specialization, closer to PM*11.1, is 2stdpc4 2354. (Contributed by BJ, 15-Sep-2018.)
|- ( A. x A. y ph -> ph )
Theorem	spsd 2057	Deduction generalizing antecedent. (Contributed by NM, 17-Aug-1994.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( A. x ps -> ch ) )
Theorem	19.2g 2058	Theorem 19.2 of [Margaris] p. 89, generalized to use two setvar variables. Use 19.2 1892 when sufficient. (Contributed by Mel L. O'Cat, 31-Mar-2008.)
|- ( A. x ph -> E. y ph )
Theorem	19.21bi 2059	Inference form of 19.21 2075 and also deduction form of sp 2053. (Contributed by NM, 26-May-1993.)
|- ( ph -> A. x ps )   =>    |- ( ph -> ps )
Theorem	19.21bbi 2060	Inference removing double quantifier. Version of 19.21bi 2059 with two quanditiers. (Contributed by NM, 20-Apr-1994.)
|- ( ph -> A. x A. y ps )   =>    |- ( ph -> ps )
Theorem	19.23bi 2061	Inference form of Theorem 19.23 of [Margaris] p. 90, see 19.23 2080. (Contributed by NM, 12-Mar-1993.)
|- ( E. x ph -> ps )   =>    |- ( ph -> ps )
Theorem	nexr 2062	Inference associated with the contrapositive of 19.8a 2052. (Contributed by Jeff Hankins, 26-Jul-2009.)
|- -. E. x ph   =>    |- -. ph
Theorem	qexmid 2063	Quantified excluded middle (see exmid 431). Also known as the drinker paradox (if ph ( x ) is interpreted as " x drinks", then this theorem tells that there exists a person such that, if this person drinks, then everyone drinks). Exercise 9.2a of Boolos, p. 111, Computability and Logic. (Contributed by NM, 10-Dec-2000.)
|- E. x ( ph -> A. x ph )
Theorem	nf5r 2064	Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.) df-nf 1710 changed. (Revised by Wolf Lammen, 11-Sep-2021.)
|- ( F/ x ph -> ( ph -> A. x ph ) )
Theorem	nf5ri 2065	Consequence of the definition of not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/ x ph   =>    |- ( ph -> A. x ph )
Theorem	nf5rd 2066	Consequence of the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- ( ph -> F/ x ps )   =>    |- ( ph -> ( ps -> A. x ps ) )
Theorem	nfim1 2067	A closed form of nfim 1825. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) df-nf 1710 changed. (Revised by Wolf Lammen, 18-Sep-2021.)
|- F/ x ph   &    |- ( ph -> F/ x ps )   =>    |- F/ x ( ph -> ps )
Theorem	nfan1 2068	A closed form of nfan 1828. (Contributed by Mario Carneiro, 3-Oct-2016.) df-nf 1710 changed. (Revised by Wolf Lammen, 18-Sep-2021.)
|- F/ x ph   &    |- ( ph -> F/ x ps )   =>    |- F/ x ( ph /\ ps )
Theorem	19.3 2069	A wff may be quantified with a variable not free in it. Theorem 19.3 of [Margaris] p. 89. See 19.3v 1897 for a version requiring fewer axioms. (Contributed by NM, 12-Mar-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
|- F/ x ph   =>    |- ( A. x ph <-> ph )
Theorem	19.9d 2070	A deduction version of one direction of 19.9 2072. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) Revised to shorten other proofs. (Revised by Wolf Lammen, 14-Jul-2020.) df-nf 1710 changed. (Revised by Wolf Lammen, 11-Sep-2021.)
|- ( ps -> F/ x ph )   =>    |- ( ps -> ( E. x ph -> ph ) )
Theorem	19.9t 2071	A closed version of 19.9 2072. (Contributed by NM, 13-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) (Proof shortened by Wolf Lammen, 14-Jul-2020.)
|- ( F/ x ph -> ( E. x ph <-> ph ) )
Theorem	19.9 2072	A wff may be existentially quantified with a variable not free in it. Theorem 19.9 of [Margaris] p. 89. See 19.9v 1896 for a version requiring fewer axioms. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) Revised to shorten other proofs. (Revised by Wolf Lammen, 14-Jul-2020.)
|- F/ x ph   =>    |- ( E. x ph <-> ph )
Theorem	19.21t 2073	Closed form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2075. (Contributed by NM, 27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 3-Jan-2018.) df-nf 1710 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof shortened by BJ, 3-Nov-2021.)
|- ( F/ x ph -> ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) ) )
Theorem	19.21tOLDOLD 2074	Obsolete proof of 19.21t 2073 as of 3-Nov-2021. (Contributed by NM, 27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 3-Jan-2018.) df-nf 1710 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( F/ x ph -> ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) ) )
Theorem	19.21 2075	Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as " x is not free in ph." See 19.21v 1868 for a version requiring fewer axioms. See also 19.21h 2121. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) df-nf 1710 changed. (Revised by Wolf Lammen, 18-Sep-2021.)
|- F/ x ph   =>    |- ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) )
Theorem	stdpc5 2076	An axiom scheme of standard predicate calculus that emulates Axiom 5 of [Mendelson] p. 69. The hypothesis F/ x ph can be thought of as emulating " x is not free in ph." With this definition, the meaning of "not free" is less restrictive than the usual textbook definition; for example x would not (for us) be free in x = x by nfequid 1940. This theorem scheme can be proved as a metatheorem of Mendelson's axiom system, even though it is slightly stronger than his Axiom 5. See stdpc5v 1867 for a version requiring fewer axioms. (Contributed by NM, 22-Sep-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 1-Jan-2018.) Remove dependency on ax-10 2019. (Revised by Wolf Lammen, 4-Jul-2021.) (Proof shortened by Wolf Lammen, 11-Oct-2021.)
|- F/ x ph   =>    |- ( A. x ( ph -> ps ) -> ( ph -> A. x ps ) )
Theorem	stdpc5OLD 2077	Obsolete proof of stdpc5 2076 as of 11-Oct-2021. (Contributed by NM, 22-Sep-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 1-Jan-2018.) Remove dependency on ax-10 2019. (Revised by Wolf Lammen, 4-Jul-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
|- F/ x ph   =>    |- ( A. x ( ph -> ps ) -> ( ph -> A. x ps ) )
Theorem	19.21-2 2078	Version of 19.21 2075 with two quantifiers. (Contributed by NM, 4-Feb-2005.)
|- F/ x ph   &    |- F/ y ph   =>    |- ( A. x A. y ( ph -> ps ) <-> ( ph -> A. x A. y ps ) )
Theorem	19.23t 2079	Closed form of Theorem 1977.23 of [Margaris] p. 90. See 19.23 2080. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) (Proof shortened by Wolf Lammen, 13-Aug-2020.) df-nf 1710 changed. (Revised by Wolf Lammen, 11-Sep-2021.)
|- ( F/ x ps -> ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) ) )
Theorem	19.23 2080	Theorem 19.23 of [Margaris] p. 90. See 19.23v 1902 for a version requiring fewer axioms. (Contributed by NM, 24-Jan-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
|- F/ x ps   =>    |- ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) )
Theorem	alimd 2081	Deduction form of Theorem 19.20 of [Margaris] p. 90, see alim 1738. (Contributed by Mario Carneiro, 24-Sep-2016.)
|- F/ x ph   &    |- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( A. x ps -> A. x ch ) )
Theorem	alrimi 2082	Inference form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2075. (Contributed by Mario Carneiro, 24-Sep-2016.)
|- F/ x ph   &    |- ( ph -> ps )   =>    |- ( ph -> A. x ps )
Theorem	alrimdd 2083	Deduction form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2075. (Contributed by Mario Carneiro, 24-Sep-2016.)
|- F/ x ph   &    |- ( ph -> F/ x ps )   &    |- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( ps -> A. x ch ) )
Theorem	alrimd 2084	Deduction form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2075. (Contributed by Mario Carneiro, 24-Sep-2016.)
|- F/ x ph   &    |- F/ x ps   &    |- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( ps -> A. x ch ) )
Theorem	eximd 2085	Deduction form of Theorem 19.22 of [Margaris] p. 90, see exim 1761. (Contributed by NM, 29-Jun-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
|- F/ x ph   &    |- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( E. x ps -> E. x ch ) )
Theorem	exlimi 2086	Inference associated with 19.23 2080. See exlimiv 1858 for a version with a dv condition requiring fewer axioms. (Contributed by NM, 10-Jan-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
|- F/ x ps   &    |- ( ph -> ps )   =>    |- ( E. x ph -> ps )
Theorem	exlimd 2087	Deduction form of Theorem 19.9 of [Margaris] p. 89. (Contributed by NM, 23-Jan-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 12-Jan-2018.)
|- F/ x ph   &    |- F/ x ch   &    |- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( E. x ps -> ch ) )
Theorem	exlimdd 2088	Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)
|- F/ x ph   &    |- F/ x ch   &    |- ( ph -> E. x ps )   &    |- ( ( ph /\ ps ) -> ch )   =>    |- ( ph -> ch )
Theorem	nexd 2089	Deduction for generalization rule for negated wff. (Contributed by Mario Carneiro, 24-Sep-2016.)
|- F/ x ph   &    |- ( ph -> -. ps )   =>    |- ( ph -> -. E. x ps )
Theorem	albid 2090	Formula-building rule for universal quantifier (deduction rule). (Contributed by Mario Carneiro, 24-Sep-2016.)
|- F/ x ph   &    |- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( A. x ps <-> A. x ch ) )
Theorem	exbid 2091	Formula-building rule for existential quantifier (deduction rule). (Contributed by Mario Carneiro, 24-Sep-2016.)
|- F/ x ph   &    |- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( E. x ps <-> E. x ch ) )
Theorem	nfbidf 2092	An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.) df-nf 1710 changed. (Revised by Wolf Lammen, 18-Sep-2021.)
|- F/ x ph   &    |- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( F/ x ps <-> F/ x ch ) )
Theorem	19.16 2093	Theorem 19.16 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
|- F/ x ph   =>    |- ( A. x ( ph <-> ps ) -> ( ph <-> A. x ps ) )
Theorem	19.17 2094	Theorem 19.17 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
|- F/ x ps   =>    |- ( A. x ( ph <-> ps ) -> ( A. x ph <-> ps ) )
Theorem	19.27 2095	Theorem 19.27 of [Margaris] p. 90. See 19.27v 1908 for a version requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)
|- F/ x ps   =>    |- ( A. x ( ph /\ ps ) <-> ( A. x ph /\ ps ) )
Theorem	19.28 2096	Theorem 19.28 of [Margaris] p. 90. See 19.28v 1909 for a version requiring fewer axioms. (Contributed by NM, 1-Aug-1993.)
|- F/ x ph   =>    |- ( A. x ( ph /\ ps ) <-> ( ph /\ A. x ps ) )
Theorem	19.19 2097	Theorem 19.19 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
|- F/ x ph   =>    |- ( A. x ( ph <-> ps ) -> ( ph <-> E. x ps ) )
Theorem	19.36 2098	Theorem 19.36 of [Margaris] p. 90. See 19.36v 1904 for a version requiring fewer axioms. (Contributed by NM, 24-Jun-1993.)
|- F/ x ps   =>    |- ( E. x ( ph -> ps ) <-> ( A. x ph -> ps ) )
Theorem	19.36i 2099	Inference associated with 19.36 2098. See 19.36iv 1905 for a version requiring fewer axioms. (Contributed by NM, 24-Jun-1993.)
|- F/ x ps   &    |- E. x ( ph -> ps )   =>    |- ( A. x ph -> ps )
Theorem	19.37 2100	Theorem 19.37 of [Margaris] p. 90. See 19.37v 1910 for a version requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)
|- F/ x ph   =>    |- ( E. x ( ph -> ps ) <-> ( ph -> E. x ps ) )