P. 327 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 32601-32700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bj-sylgt2 32601	Uncurried (imported) form of sylgt 1749. (Contributed by BJ, 2-May-2019.)
|- ( ( A. x ( ps -> ch ) /\ ( ph -> A. x ps ) ) -> ( ph -> A. x ch ) )
Theorem	bj-exlimh 32602	Closed form of close to exlimih 2148. (Contributed by BJ, 2-May-2019.)
|- ( A. x ( ph -> ps ) -> ( ( E. x ps -> ch ) -> ( E. x ph -> ch ) ) )
Theorem	bj-exlimh2 32603	Uncurried (imported) form of bj-exlimh 32602. (Contributed by BJ, 2-May-2019.)
|- ( ( A. x ( ph -> ps ) /\ ( E. x ps -> ch ) ) -> ( E. x ph -> ch ) )
Theorem	bj-alrimhi 32604	An inference associated with sylgt 1749 and bj-exlimh 32602. (Contributed by BJ, 12-May-2019.)
|- ( ph -> ps )   =>    |- ( F/ x ph -> ( E. x ph -> A. x ps ) )
Theorem	bj-alexim 32605	Closed form of aleximi 1759 (with a shorter proof, so aleximi 1759 could be deduced from it -- exim 1761 would have to be proved first, but its proof is shorter (currently almost a subproof of aleximi 1759)). (Contributed by BJ, 8-Nov-2021.)
|- ( A. x ( ph -> ( ps -> ch ) ) -> ( A. x ph -> ( E. x ps -> E. x ch ) ) )
Theorem	bj-nexdh 32606	Closed form of nexdh 1792 (actually, its general instance). (Contributed by BJ, 6-May-2019.)
|- ( A. x ( ph -> -. ps ) -> ( ( ch -> A. x ph ) -> ( ch -> -. E. x ps ) ) )
Theorem	bj-nexdh2 32607	Uncurried (imported) form of bj-nexdh 32606. (Contributed by BJ, 6-May-2019.)
|- ( ( A. x ( ph -> -. ps ) /\ ( ch -> A. x ph ) ) -> ( ch -> -. E. x ps ) )
Theorem	bj-hbxfrbi 32608	Closed form of hbxfrbi 1752. Notes: it is less important than nfbiit 1777; it requires sp 2053 (unlike nfbiit 1777); there is an obvious version with ( E. x ph -> ph ) instead. (Contributed by BJ, 6-May-2019.)
|- ( A. x ( ph <-> ps ) -> ( ( ph -> A. x ph ) <-> ( ps -> A. x ps ) ) )
Theorem	bj-exlime 32609	Variant of exlimih 2148 where the non-freeness of x in ps is expressed using an existential quantifier, thus requiring fewer axioms. (Contributed by BJ, 17-Mar-2020.)
|- ( E. x ps -> ps )   &    |- ( ph -> ps )   =>    |- ( E. x ph -> ps )
Theorem	bj-exnalimn 32610	A transformation of quantifiers and logical connectives. The general statement that equs3 1875 proves. This and the following theorems are the general instances of already proved theorems. They could be moved to the main part, before ax-5 1839. I propose to move to the main part: bj-exnalimn 32610, bj-exalim 32611, bj-exalimi 32612, bj-exalims 32613, bj-exalimsi 32614, bj-ax12i 32616, bj-ax12wlem 32617, bj-ax12w 32665, and remove equs3 1875. A new label is needed for bj-ax12i 32616 and label suggestions are welcome for the others. I also propose to change -. A. x -. to E. x in speimfw 1876 and spimfw 1878 (other spim* theorems use E. x and very few theorems in set.mm use -. A. x -.). (Contributed by BJ, 29-Sep-2019.)
|- ( E. x ( ph /\ ps ) <-> -. A. x ( ph -> -. ps ) )
Theorem	bj-exalim 32611	Distributing quantifiers over a double implication. (Contributed by BJ, 8-Nov-2021.)
|- ( A. x ( ph -> ( ps -> ch ) ) -> ( E. x ph -> ( A. x ps -> E. x ch ) ) )
Theorem	bj-exalimi 32612	An inference for distributing quantifiers over a double implication. (Almost) the general statement that speimfw 1876 proves. (Contributed by BJ, 29-Sep-2019.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( E. x ph -> ( A. x ps -> E. x ch ) )
Theorem	bj-exalims 32613	Distributing quantifiers over a double implication. (Almost) the general statement that spimfw 1878 proves. (Contributed by BJ, 29-Sep-2019.)
|- ( E. x ph -> ( -. ch -> A. x -. ch ) )   =>    |- ( A. x ( ph -> ( ps -> ch ) ) -> ( E. x ph -> ( A. x ps -> ch ) ) )
Theorem	bj-exalimsi 32614	An inference for distributing quantifiers over a double implication. (Almost) the general statement that spimfw 1878 proves. (Contributed by BJ, 29-Sep-2019.)
|- ( ph -> ( ps -> ch ) )   &    |- ( E. x ph -> ( -. ch -> A. x -. ch ) )   =>    |- ( E. x ph -> ( A. x ps -> ch ) )
Theorem	bj-ax12ig 32615	A lemma used to prove a weak form of the axiom of substitution. A generalization of bj-ax12i 32616. (Contributed by BJ, 19-Dec-2020.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ph -> ( ch -> A. x ch ) )   =>    |- ( ph -> ( ps -> A. x ( ph -> ps ) ) )
Theorem	bj-ax12i 32616	A weakening of bj-ax12ig 32615 that is sufficient to prove a weak form of the axiom of substitution ax-12 2047. The general statement of which ax12i 1879 is an instance. (Contributed by BJ, 29-Sep-2019.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ch -> A. x ch )   =>    |- ( ph -> ( ps -> A. x ( ph -> ps ) ) )
20.14.4.3  Adding ax-5
Theorem	bj-ax12wlem 32617*	A lemma used to prove a weak version of the axiom of substitution ax-12 2047. (Temporary comment: The general statement that ax12wlem 2009 proves.) (Contributed by BJ, 20-Mar-2020.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( ps -> A. x ( ph -> ps ) ) )
20.14.4.4  Equality and substitution
Theorem	bj-ssbjust 32618*	Justification theorem for df-ssb 32620 from Tarski's FOL. (Contributed by BJ, 9-Nov-2021.)
|- ( A. y ( y = t -> A. x ( x = y -> ph ) ) <-> A. z ( z = t -> A. x ( x = z -> ph ) ) )
Syntax	wssb 32619	Syntax for the substitution of a variable for a variable in a formula. (Contributed by BJ, 22-Dec-2020.)
wff [ t/ x]b ph
Definition	df-ssb 32620*	Alternate definition of proper substitution. Note that the occurrences of a given variable are either all bound ( x , y) or all free ( t). Also note that the definiens uses only primitive symbols. It is obtained by applying twice Tarski's definition sb6 2429 which is valid for disjoint variables, so we introduce a dummy variable y to isolate x from t, as in dfsb7 2455 with respect to sb5 2430. This double level definition will make several proofs using it appear as doubled. Alternately, one could often first prove as a lemma the same theorem with a DV condition on the substitute and the substituted variables, and then prove the original theorem by applying this lemma twice in a row. This definition uses a dummy variable, but the justification theorem, bj-ssbjust 32618, is provable from Tarski's FOL. Once this is proved, more of the fundamental properties of proper substitution will be provable from Tarski's FOL system, sometimes with the help of specialization sp 2053, of the substitution axiom ax-12 2047, and of commutation of quantifiers ax-11 2034; that is, ax-13 2246 will often be avoided. (Contributed by BJ, 22-Dec-2020.)
|- ([ t/ x]b ph <-> A. y ( y = t -> A. x ( x = y -> ph ) ) )
Theorem	bj-ssbim 32621	Distribute substitution over implication, closed form. Specialization of implication. Uses only ax-1--5. Compare spsbim 2394. (Contributed by BJ, 22-Dec-2020.)
|- ( A. x ( ph -> ps ) -> ([ t/ x]b ph -> [ t/ x]b ps ) )
Theorem	bj-ssbbi 32622	Biconditional property for substitution, closed form. Specialization of biconditional. Uses only ax-1--5. Compare spsbbi 2402. (Contributed by BJ, 22-Dec-2020.)
|- ( A. x ( ph <-> ps ) -> ([ t/ x]b ph <-> [ t/ x]b ps ) )
Theorem	bj-ssbimi 32623	Distribute substitution over implication. Uses only ax-1--5. (Contributed by BJ, 22-Dec-2020.)
|- ( ph -> ps )   =>    |- ([ t/ x]b ph -> [ t/ x]b ps )
Theorem	bj-ssbbii 32624	Biconditional property for substitution. Uses only ax-1--5. (Contributed by BJ, 22-Dec-2020.)
|- ( ph <-> ps )   =>    |- ([ t/ x]b ph <-> [ t/ x]b ps )
Theorem	bj-alsb 32625	If a proposition is true for all instances, then it is true for any specific one. Uses only ax-1--5. Compare stdpc4 2353 which uses auxiliary axioms. (Contributed by BJ, 22-Dec-2020.)
|- ( A. x ph -> [ t/ x]b ph )
Theorem	bj-sbex 32626	If a proposition is true for a specific instance, then there exists an instance such that it is true for it. Uses only ax-1--6. Compare spsbe 1884 which, due to the specific form of df-sb 1881, uses fewer axioms. (Contributed by BJ, 22-Dec-2020.)
|- ([ t/ x]b ph -> E. x ph )
Theorem	bj-ssbeq 32627*	Substitution in an equality, disjoint variables case. Uses only ax-1--6. It might be shorter to prove the result about composition of two substitutions and prove bj-ssbeq 32627 first with a DV on x,t, and then in the general case. (Contributed by BJ, 22-Dec-2020.)
|- ([ t/ x]b y = z <-> y = z )
Theorem	bj-ssb0 32628*	Substitution for a variable not occurring in a proposition. See sbf 2380. (Contributed by BJ, 22-Dec-2020.)
|- ([ t/ x]b ph <-> ph )
Theorem	bj-ssbequ 32629	Equality property for substitution, from Tarski's system. Compare sbequ 2376. (Contributed by BJ, 30-Dec-2020.)
|- ( s = t -> ([ s/ x]b ph <-> [ t/ x]b ph ) )
Theorem	bj-ssblem1 32630*	A lemma for the definiens of df-sb 1881. An instance of sp 2053 proved without it. Note: it has a common subproof with bj-ssbjust 32618. (Contributed by BJ, 22-Dec-2020.)
|- ( A. y ( y = t -> A. x ( x = y -> ph ) ) -> ( y = t -> A. x ( x = y -> ph ) ) )
Theorem	bj-ssblem2 32631*	An instance of ax-11 2034 proved without it. The converse may not be provable without ax-11 2034 (since using alcomiw 1971 would require a DV on ph , x, which defeats the purpose). (Contributed by BJ, 22-Dec-2020.)
|- ( A. x A. y ( y = t -> ( x = y -> ph ) ) -> A. y A. x ( y = t -> ( x = y -> ph ) ) )
Theorem	bj-ssb1a 32632*	One direction of a simplified definition of substitution in case of disjoint variables. See bj-ssb1 32633 for the biconditional, which requires ax-11 2034. (Contributed by BJ, 22-Dec-2020.)
|- ( A. x ( x = t -> ph ) -> [ t/ x]b ph )
Theorem	bj-ssb1 32633*	A simplified definition of substitution in case of disjoint variables. See bj-ssb1a 32632 for the backward implication, which does not require ax-11 2034 (note that here, the version of ax-11 2034 with disjoint setvar metavariables would suffice). Compare sb6 2429. (Contributed by BJ, 22-Dec-2020.)
|- ([ t/ x]b ph <-> A. x ( x = t -> ph ) )
Theorem	bj-ax12 32634*	A weaker form of ax-12 2047 and ax12v2 2049, namely the generalization over x of the latter. In this statement, all occurrences of x are bound. (Contributed by BJ, 26-Dec-2020.)
|- A. x ( x = t -> ( ph -> A. x ( x = t -> ph ) ) )
Theorem	bj-ax12ssb 32635*	The axiom bj-ax12 32634 expressed using substitution. (Contributed by BJ, 26-Dec-2020.)
|- [ t/ x]b ( ph -> [ t/ x]b ph )
Theorem	bj-modal5e 32636	Dual statement of hbe1 2021 (which is the real modal-5 2032). See also axc7 2132 and axc7e 2133. (Contributed by BJ, 21-Dec-2020.)
|- ( E. x A. x ph -> A. x ph )
Theorem	bj-19.41al 32637	Special case of 19.41 2103 proved from Tarski, ax-10 2019 (modal5) and hba1 2151 (modal4). (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
|- ( E. x ( ph /\ A. x ps ) <-> ( E. x ph /\ A. x ps ) )
Theorem	bj-equsexval 32638*	Special case of equsexv 2109 proved from Tarski, ax-10 2019 (modal5) and hba1 2151 (modal4). (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
|- ( x = y -> ( ph <-> A. x ps ) )   =>    |- ( E. x ( x = y /\ ph ) <-> A. x ps )
Theorem	bj-sb56 32639*	Proof of sb56 2150 from Tarski, ax-10 2019 (modal5) and bj-ax12 32634. (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
|- ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) )
Theorem	bj-dfssb2 32640*	An alternate definition of df-ssb 32620. Note that the use of a dummy variable in the definition df-ssb 32620 allows to use bj-sb56 32639 instead of equs45f 2350 and hence to avoid dependency on ax-13 2246 and to use ax-12 2047 only through bj-ax12 32634. Compare dfsb7 2455. (Contributed by BJ, 25-Dec-2020.)
|- ([ t/ x]b ph <-> E. y ( y = t /\ E. x ( x = y /\ ph ) ) )
Theorem	bj-ssbn 32641	The result of a substitution in the negation of a formula is the negation of the result of the same substitution in that formula. Proved from Tarski, ax-10 2019, bj-ax12 32634. Compare sbn 2391. (Contributed by BJ, 25-Dec-2020.)
|- ([ t/ x]b -. ph <-> -. [ t/ x]b ph )
Theorem	bj-ssbft 32642	See sbft 2379. This proof is from Tarski's FOL together with sp 2053 (and its dual). (Contributed by BJ, 22-Dec-2020.)
|- ( F/ x ph -> ([ t/ x]b ph <-> ph ) )
Theorem	bj-ssbequ2 32643	Note that ax-12 2047 is used only via sp 2053. See sbequ2 1882 and stdpc7 1958. (Contributed by BJ, 22-Dec-2020.)
|- ( x = t -> ([ t/ x]b ph -> ph ) )
Theorem	bj-ssbequ1 32644	This uses ax-12 2047 with a direct reference to ax12v 2048. Therefore, compared to bj-ax12 32634, there is a hidden use of sp 2053. Note that with ax-12 2047, it can be proved with dv condition on x , t. See sbequ1 2110. (Contributed by BJ, 22-Dec-2020.)
|- ( x = t -> ( ph -> [ t/ x]b ph ) )
Theorem	bj-ssbid2 32645	A special case of bj-ssbequ2 32643. (Contributed by BJ, 22-Dec-2020.)
|- ([ x/ x]b ph -> ph )
Theorem	bj-ssbid2ALT 32646	Alternate proof of bj-ssbid2 32645, not using bj-ssbequ2 32643. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ([ x/ x]b ph -> ph )
Theorem	bj-ssbid1 32647	A special case of bj-ssbequ1 32644. (Contributed by BJ, 22-Dec-2020.)
|- ( ph -> [ x/ x]b ph )
Theorem	bj-ssbid1ALT 32648	Alternate proof of bj-ssbid1 32647, not using bj-ssbequ1 32644. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ph -> [ x/ x]b ph )
Theorem	bj-ssbssblem 32649*	Composition of two substitutions with a fresh intermediate variable. Remark: does not seem useful. (Contributed by BJ, 22-Dec-2020.)
|- ([ t/ y]b[ y/ x]b ph <-> [ t/ x]b ph )
Theorem	bj-ssbcom3lem 32650*	Lemma for bj-ssbcom3 when setvar variables are disjoint. Remark: does not seem useful. (Contributed by BJ, 30-Dec-2020.)
|- ([ t/ y]b[ y/ x]b ph <-> [ t/ y]b[ t/ x]b ph )
Theorem	bj-ax6elem1 32651*	Lemma for bj-ax6e 32653. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.)
|- ( -. A. x x = y -> ( y = z -> A. x y = z ) )
Theorem	bj-ax6elem2 32652*	Lemma for bj-ax6e 32653. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.)
|- ( A. x y = z -> E. x x = y )
Theorem	bj-ax6e 32653	Proof of ax6e 2250 (hence ax6 2251) from Tarski's system, ax-c9 34175, ax-c16 34177. Remark: ax-6 1888 is used only via its principal (unbundled) instance ax6v 1889. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
|- E. x x = y
20.14.4.5  Adding ax-6
Theorem	bj-extru 32654	There exists a variable such that T. holds; that is, there exists a variable. This corresponds under the standard translation to one of the formulations of the modal axiom (D), the other being 19.2 1892. (This is also extt 32403; propose to move to Main extt 32403 and allt 32400; relabel exiftru 1891 to "exgen", for "existential generalization", which is the standard name for that rule of inference ? ). (Contributed by BJ, 12-May-2019.) (Proof modification is discouraged.)
|- E. x T.
Theorem	bj-alequexv 32655*	Version of bj-alequex 32708 with DV(x,y), requiring fewer axioms. (Contributed by BJ, 9-Nov-2021.)
|- ( A. x ( x = y -> ph ) -> E. x ph )
Theorem	bj-spimvwt 32656*	Closed form of spimvw 1927. See also spimt 2253. (Contributed by BJ, 8-Nov-2021.)
|- ( A. x ( x = y -> ( ph -> ps ) ) -> ( A. x ph -> ps ) )
Theorem	bj-spimevw 32657*	Existential introduction, using implicit substitution. This is to spimeh 1925 what spimvw 1927 is to spimw 1926. (Contributed by BJ, 17-Mar-2020.)
|- ( x = y -> ( ph -> ps ) )   =>    |- ( ph -> E. x ps )
Theorem	bj-spnfw 32658	Theorem close to a closed form of spnfw 1928. (Contributed by BJ, 12-May-2019.)
|- ( ( E. x ph -> ps ) -> ( A. x ph -> ps ) )
Theorem	bj-cbvexiw 32659*	Change bound variable. This is to cbvexvw 1970 what cbvaliw 1933 is to cbvalvw 1969. [TODO: move after cbvalivw 1934]. (Contributed by BJ, 17-Mar-2020.)
|- ( E. x E. y ps -> E. y ps )   &    |- ( ph -> A. y ph )   &    |- ( y = x -> ( ph -> ps ) )   =>    |- ( E. x ph -> E. y ps )
Theorem	bj-cbvexivw 32660*	Change bound variable. This is to cbvexvw 1970 what cbvalivw 1934 is to cbvalvw 1969. [TODO: move after cbvalivw 1934]. (Contributed by BJ, 17-Mar-2020.)
|- ( y = x -> ( ph -> ps ) )   =>    |- ( E. x ph -> E. y ps )
Theorem	bj-modald 32661	A short form of the axiom D of modal logic. (Contributed by BJ, 4-Apr-2021.)
|- ( A. x -. ph -> -. A. x ph )
Theorem	bj-denot 32662*	A weakening of ax-6 1888 and ax6v 1889. (Contributed by BJ, 4-Apr-2021.) (New usage is discouraged.)
|- ( x = x -> -. A. y -. y = x )
Theorem	bj-eqs 32663*	A lemma for substitutions, proved from Tarski's FOL. The version without DV( x, y) is true but requires ax-13 2246. The DV condition DV( x, ph) is necessary for both directions: consider substituting x = z for ph. (Contributed by BJ, 25-May-2021.)
|- ( ph <-> A. x ( x = y -> ph ) )
20.14.4.6  Adding ax-7
Theorem	bj-cbvexw 32664*	Change bound variable. This is to cbvexvw 1970 what cbvalw 1968 is to cbvalvw 1969. (Contributed by BJ, 17-Mar-2020.)
|- ( E. x E. y ps -> E. y ps )   &    |- ( ph -> A. y ph )   &    |- ( E. y E. x ph -> E. x ph )   &    |- ( ps -> A. x ps )   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( E. x ph <-> E. y ps )
Theorem	bj-ax12w 32665*	The general statement that ax12w 2010 proves. (Contributed by BJ, 20-Mar-2020.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( y = z -> ( ps <-> th ) )   =>    |- ( ph -> ( A. y ps -> A. x ( ph -> ps ) ) )
20.14.4.7  Membership predicate, ax-8 and ax-9
Theorem	bj-elequ2g 32666*	A form of elequ2 2004 with a universal quantifier. Its converse is ax-ext 2602. (TODO: move to main part, minimize axext4 2606--- as of 4-Nov-2020, minimizes only axext4 2606, by 13 bytes; and link to it in the comment of ax-ext 2602.) (Contributed by BJ, 3-Oct-2019.)
|- ( x = y -> A. z ( z e. x <-> z e. y ) )
Theorem	bj-ax89 32667	A theorem which could be used as sole axiom for the non-logical predicate instead of ax-8 1992 and ax-9 1999. Indeed, it is implied over propositional calculus by the conjunction of ax-8 1992 and ax-9 1999, as proved here. In the other direction, one can prove ax-8 1992 (respectively ax-9 1999) from bj-ax89 32667 by using mpan2 707 ( respectively mpan 706) and equid 1939. (TODO: move to main part.) (Contributed by BJ, 3-Oct-2019.)
|- ( ( x = y /\ z = t ) -> ( x e. z -> y e. t ) )
Theorem	bj-elequ12 32668	An identity law for the non-logical predicate, which combines elequ1 1997 and elequ2 2004. For the analogous theorems for class terms, see eleq1 2689, eleq2 2690 and eleq12 2691. (TODO: move to main part.) (Contributed by BJ, 29-Sep-2019.)
|- ( ( x = y /\ z = t ) -> ( x e. z <-> y e. t ) )
Theorem	bj-cleljusti 32669*	One direction of cleljust 1998, requiring only ax-1 6-- ax-5 1839 and ax8v1 1994. (Contributed by BJ, 31-Dec-2020.) (Proof modification is discouraged.)
|- ( E. z ( z = x /\ z e. y ) -> x e. y )
20.14.4.8  Adding ax-11
Theorem	bj-alcomexcom 32670	Commutation of universal quantifiers implies commutation of existential quantifiers. Can be placed in the ax-4 1737 section, soon after 2nexaln 1757, and used to prove excom 2042. (Contributed by BJ, 29-Nov-2020.) (Proof modification is discouraged.)
|- ( ( A. x A. y -. ph -> A. y A. x -. ph ) -> ( E. y E. x ph -> E. x E. y ph ) )
Theorem	bj-hbalt 32671	Closed form of hbal 2036. When in main part, prove hbal 2036 and hbald 2041 from it. (Contributed by BJ, 2-May-2019.)
|- ( A. y ( ph -> A. x ph ) -> ( A. y ph -> A. x A. y ph ) )
20.14.4.9  Adding ax-12
Theorem	axc11n11 32672	Proof of axc11n 2307 from { ax-1 6-- ax-7 1935, axc11 2314 } . Almost identical to axc11nfromc11 34211. (Contributed by NM, 6-Jul-2021.) (Proof modification is discouraged.)
|- ( A. x x = y -> A. y y = x )
Theorem	axc11n11r 32673	Proof of axc11n 2307 from { ax-1 6-- ax-7 1935, axc9 2302, axc11r 2187 } (note that axc16 2135 is provable from { ax-1 6-- ax-7 1935, axc11r 2187 }). Note that axc11n 2307 proves (over minimal calculus) that axc11 2314 and axc11r 2187 are equivalent. Therefore, axc11n11 32672 and axc11n11r 32673 prove that one can use one or the other as an axiom, provided one assumes the axioms listed above (axc11 2314 appears slightly stronger since axc11n11r 32673 requires axc9 2302 while axc11n11 32672 does not). (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.)
|- ( A. x x = y -> A. y y = x )
Theorem	bj-axc16g16 32674*	Proof of axc16g 2134 from { ax-1 6-- ax-7 1935, axc16 2135 }. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.)
|- ( A. x x = y -> ( ph -> A. z ph ) )
Theorem	bj-ax12v3 32675*	A weak version of ax-12 2047 which is stronger than ax12v 2048. Note that if one assumes reflexivity of equality |- x = x (equid 1939), then bj-ax12v3 32675 implies ax-5 1839 over modal logic K (substitute x for y). See also bj-ax12v3ALT 32676. (Contributed by BJ, 6-Jul-2021.)
|- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) )
Theorem	bj-ax12v3ALT 32676*	Alternate proof of bj-ax12v3 32675. Uses axc11r 2187 and axc15 2303 instead of ax-12 2047. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) )
Theorem	bj-sb 32677*	A weak variant of sbid2 2413 not requiring ax-13 2246 nor ax-10 2019. On top of Tarski's FOL, one implication requires only ax12v 2048, and the other requires only sp 2053. (Contributed by BJ, 25-May-2021.)
|- ( ph <-> A. y ( y = x -> A. x ( x = y -> ph ) ) )
Theorem	bj-modalbe 32678	The predicate-calculus version of the axiom (B) of modal logic. See also modal-b 2142. (Contributed by BJ, 20-Oct-2019.)
|- ( ph -> A. x E. x ph )
Theorem	bj-spst 32679	Closed form of sps 2055. Once in main part, prove sps 2055 and spsd 2057 from it. (Contributed by BJ, 20-Oct-2019.)
|- ( ( ph -> ps ) -> ( A. x ph -> ps ) )
Theorem	bj-19.21bit 32680	Closed form of 19.21bi 2059. (Contributed by BJ, 20-Oct-2019.)
|- ( ( ph -> A. x ps ) -> ( ph -> ps ) )
Theorem	bj-19.23bit 32681	Closed form of 19.23bi 2061. (Contributed by BJ, 20-Oct-2019.)
|- ( ( E. x ph -> ps ) -> ( ph -> ps ) )
Theorem	bj-nexrt 32682	Closed form of nexr 2062. Contrapositive of 19.8a 2052. (Contributed by BJ, 20-Oct-2019.)
|- ( -. E. x ph -> -. ph )
Theorem	bj-alrim 32683	Closed form of alrimi 2082. (Contributed by BJ, 2-May-2019.)
|- ( F/ x ph -> ( A. x ( ph -> ps ) -> ( ph -> A. x ps ) ) )
Theorem	bj-alrim2 32684	Uncurried (imported) form of bj-alrim 32683. (Contributed by BJ, 2-May-2019.)
|- ( ( F/ x ph /\ A. x ( ph -> ps ) ) -> ( ph -> A. x ps ) )
Theorem	bj-nfdt0 32685	A theorem close to a closed form of nf5d 2118 and nf5dh 2026. (Contributed by BJ, 2-May-2019.)
|- ( A. x ( ph -> ( ps -> A. x ps ) ) -> ( A. x ph -> F/ x ps ) )
Theorem	bj-nfdt 32686	Closed form of nf5d 2118 and nf5dh 2026. (Contributed by BJ, 2-May-2019.)
|- ( A. x ( ph -> ( ps -> A. x ps ) ) -> ( ( ph -> A. x ph ) -> ( ph -> F/ x ps ) ) )
Theorem	bj-nexdt 32687	Closed form of nexd 2089. (Contributed by BJ, 20-Oct-2019.)
|- ( F/ x ph -> ( A. x ( ph -> -. ps ) -> ( ph -> -. E. x ps ) ) )
Theorem	bj-nexdvt 32688*	Closed form of nexdv 1864. (Contributed by BJ, 20-Oct-2019.)
|- ( A. x ( ph -> -. ps ) -> ( ph -> -. E. x ps ) )
Theorem	bj-19.3t 32689	Closed form of 19.3 2069. (Contributed by BJ, 20-Oct-2019.)
|- ( ( ph -> A. x ph ) -> ( A. x ph <-> ph ) )
Theorem	bj-alexbiex 32690	Adding a second quantifier is a tranparent operation, ( A. E. case). (Contributed by BJ, 20-Oct-2019.)
|- ( A. x E. x ph <-> E. x ph )
Theorem	bj-exexbiex 32691	Adding a second quantifier is a tranparent operation, ( E. E. case). (Contributed by BJ, 20-Oct-2019.)
|- ( E. x E. x ph <-> E. x ph )
Theorem	bj-alalbial 32692	Adding a second quantifier is a tranparent operation, ( A. A. case). (Contributed by BJ, 20-Oct-2019.)
|- ( A. x A. x ph <-> A. x ph )
Theorem	bj-exalbial 32693	Adding a second quantifier is a tranparent operation, ( E. A. case). (Contributed by BJ, 20-Oct-2019.)
|- ( E. x A. x ph <-> A. x ph )
Theorem	bj-19.9htbi 32694	Strengthening 19.9ht 2143 by replacing its succedent with a biconditional (19.9t 2071 does have a biconditional succedent). This propagates. (Contributed by BJ, 20-Oct-2019.)
|- ( A. x ( ph -> A. x ph ) -> ( E. x ph <-> ph ) )
Theorem	bj-hbntbi 32695	Strengthening hbnt 2144 by replacing its succedent with a biconditional. See also hbntg 31711 and hbntal 38769. (Contributed by BJ, 20-Oct-2019.) Proved from bj-19.9htbi 32694. (Proof modification is discouraged.)
|- ( A. x ( ph -> A. x ph ) -> ( -. ph <-> A. x -. ph ) )
Theorem	bj-biexal1 32696	A general FOL biconditional that generalizes 19.9ht 2143 among others. For this and the following theorems, see also 19.35 1805, 19.21 2075, 19.23 2080. (Contributed by BJ, 20-Oct-2019.)
|- ( A. x ( ph -> A. x ps ) <-> ( E. x ph -> A. x ps ) )
Theorem	bj-biexal2 32697	A general FOL biconditional. (Contributed by BJ, 20-Oct-2019.)
|- ( A. x ( E. x ph -> ps ) <-> ( E. x ph -> A. x ps ) )
Theorem	bj-biexal3 32698	A general FOL biconditional. (Contributed by BJ, 20-Oct-2019.)
|- ( A. x ( ph -> A. x ps ) <-> A. x ( E. x ph -> ps ) )
Theorem	bj-bialal 32699	A general FOL biconditional. (Contributed by BJ, 20-Oct-2019.)
|- ( A. x ( A. x ph -> ps ) <-> ( A. x ph -> A. x ps ) )
Theorem	bj-biexex 32700	A general FOL biconditional. (Contributed by BJ, 20-Oct-2019.)
|- ( A. x ( ph -> E. x ps ) <-> ( E. x ph -> E. x ps ) )