P. 35 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 3401-3500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	reu7 3401*	Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( E! x e. A ph <-> ( E. x e. A ph /\ E. x e. A A. y e. A ( ps -> x = y ) ) )
Theorem	reu8 3402*	Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( E! x e. A ph <-> E. x e. A ( ph /\ A. y e. A ( ps -> x = y ) ) )
Theorem	reu2eqd 3403*	Deduce equality from restricted uniqueness, deduction version. (Contributed by Thierry Arnoux, 27-Nov-2019.)
|- ( x = B -> ( ps <-> ch ) )   &    |- ( x = C -> ( ps <-> th ) )   &    |- ( ph -> E! x e. A ps )   &    |- ( ph -> B e. A )   &    |- ( ph -> C e. A )   &    |- ( ph -> ch )   &    |- ( ph -> th )   =>    |- ( ph -> B = C )
Theorem	reueq 3404*	Equality has existential uniqueness. (Contributed by Mario Carneiro, 1-Sep-2015.)
|- ( B e. A <-> E! x e. A x = B )
Theorem	rmoeq 3405*	Equality's restricted existential "at most one" property. (Contributed by Thierry Arnoux, 30-Mar-2018.) (Revised by AV, 27-Oct-2020.) (Proof shortened by NM, 29-Oct-2020.)
|- E* x e. B x = A
Theorem	rmoan 3406	Restricted "at most one" still holds when a conjunct is added. (Contributed by NM, 16-Jun-2017.)
|- ( E* x e. A ph -> E* x e. A ( ps /\ ph ) )
Theorem	rmoim 3407	Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.)
|- ( A. x e. A ( ph -> ps ) -> ( E* x e. A ps -> E* x e. A ph ) )
Theorem	rmoimia 3408	Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.)
|- ( x e. A -> ( ph -> ps ) )   =>    |- ( E* x e. A ps -> E* x e. A ph )
Theorem	rmoimi2 3409	Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.)
|- A. x ( ( x e. A /\ ph ) -> ( x e. B /\ ps ) )   =>    |- ( E* x e. B ps -> E* x e. A ph )
Theorem	2reuswap 3410*	A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by NM, 16-Jun-2017.)
|- ( A. x e. A E* y e. B ph -> ( E! x e. A E. y e. B ph -> E! y e. B E. x e. A ph ) )
Theorem	reuind 3411*	Existential uniqueness via an indirect equality. (Contributed by NM, 16-Oct-2010.)
|- ( x = y -> ( ph <-> ps ) )   &    |- ( x = y -> A = B )   =>    |- ( ( A. x A. y ( ( ( A e. C /\ ph ) /\ ( B e. C /\ ps ) ) -> A = B ) /\ E. x ( A e. C /\ ph ) ) -> E! z e. C A. x ( ( A e. C /\ ph ) -> z = A ) )
Theorem	2rmorex 3412*	Double restricted quantification with "at most one," analogous to 2moex 2543. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
|- ( E* x e. A E. y e. B ph -> A. y e. B E* x e. A ph )
Theorem	2reu5lem1 3413*	Lemma for 2reu5 3416. Note that E! x e. A E! y e. B ph does not mean "there is exactly one x in A and exactly one y in B such that ph holds;" see comment for 2eu5 2557. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
|- ( E! x e. A E! y e. B ph <-> E! x E! y ( x e. A /\ y e. B /\ ph ) )
Theorem	2reu5lem2 3414*	Lemma for 2reu5 3416. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
|- ( A. x e. A E* y e. B ph <-> A. x E* y ( x e. A /\ y e. B /\ ph ) )
Theorem	2reu5lem3 3415*	Lemma for 2reu5 3416. This lemma is interesting in its own right, showing that existential restriction in the last conjunct (the "at most one" part) is optional; compare rmo2 3526. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
|- ( ( E! x e. A E! y e. B ph /\ A. x e. A E* y e. B ph ) <-> ( E. x e. A E. y e. B ph /\ E. z E. w A. x e. A A. y e. B ( ph -> ( x = z /\ y = w ) ) ) )
Theorem	2reu5 3416*	Double restricted existential uniqueness in terms of restricted existential quantification and restricted universal quantification, analogous to 2eu5 2557 and reu3 3396. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
|- ( ( E! x e. A E! y e. B ph /\ A. x e. A E* y e. B ph ) <-> ( E. x e. A E. y e. B ph /\ E. z e. A E. w e. B A. x e. A A. y e. B ( ph -> ( x = z /\ y = w ) ) ) )
Theorem	nelrdva 3417*	Deduce negative membership from an implication. (Contributed by Thierry Arnoux, 27-Nov-2017.)
|- ( ( ph /\ x e. A ) -> x =/= B )   =>    |- ( ph -> -. B e. A )
2.1.7  Conditional equality (experimental) This is a very useless definition, which "abbreviates" ( x = y -> ph ) as CondEq ( x = y -> ph ). What this display hides, though, is that the first expression, even though it has a shorter constant string, is actually much more complicated in its parse tree: it is parsed as (wi (wceq (cv vx) (cv vy)) wph), while the CondEq version is parsed as (wcdeq vx vy wph). It also allows us to give a name to the specific ternary operation ( x = y -> ph ). This is all used as part of a metatheorem: we want to say that |- ( x = y -> ( ph ( x ) <-> ph ( y ) ) ) and |- ( x = y -> A ( x ) = A ( y ) ) are provable, for any expressions ph ( x ) or A ( x ) in the language. The proof is by induction, so the base case is each of the primitives, which is why you will see a theorem for each of the set.mm primitive operations. The metatheorem comes with a disjoint variables assumption: every variable in ph ( x ) is assumed disjoint from x except x itself. For such a proof by induction, we must consider each of the possible forms of ph ( x ). If it is a variable other than x, then we have CondEq ( x = y -> A = A ) or CondEq ( x = y -> ( ph <-> ph ) ), which is provable by cdeqth 3422 and reflexivity. Since we are only working with class and wff expressions, it can't be x itself in set.mm, but if it was we'd have to also prove CondEq ( x = y -> x = y ) (where set equality is being used on the right). Otherwise, it is a primitive operation applied to smaller expressions. In these cases, for each setvar variable parameter to the operation, we must consider if it is equal to x or not, which yields 2^n proof obligations. Luckily, all primitive operations in set.mm have either zero or one setvar variable, so we only need to prove one statement for the non-set constructors (like implication) and two for the constructors taking a set (the forall and the class builder). In each of the primitive proofs, we are allowed to assume that y is disjoint from ph ( x ) and vice versa, because this is maintained through the induction. This is how we satisfy the dv conditions of cdeqab1 3427 and cdeqab 3425.
Syntax	wcdeq 3418	Extend wff notation to include conditional equality. This is a technical device used in the proof that F/ is the not-free predicate, and that definitions are conservative as a result.
wff CondEq ( x = y -> ph )
Definition	df-cdeq 3419	Define conditional equality. All the notation to the left of the <-> is fake; the parentheses and arrows are all part of the notation, which could equally well be written CondEq x y ph. On the right side is the actual implication arrow. The reason for this definition is to "flatten" the structure on the right side (whose tree structure is something like (wi (wceq (cv vx) (cv vy)) wph) ) into just (wcdeq vx vy wph). (Contributed by Mario Carneiro, 11-Aug-2016.)
|- (CondEq ( x = y -> ph ) <-> ( x = y -> ph ) )
Theorem	cdeqi 3420	Deduce conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- ( x = y -> ph )   =>    |- CondEq ( x = y -> ph )
Theorem	cdeqri 3421	Property of conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- CondEq ( x = y -> ph )   =>    |- ( x = y -> ph )
Theorem	cdeqth 3422	Deduce conditional equality from a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- ph   =>    |- CondEq ( x = y -> ph )
Theorem	cdeqnot 3423	Distribute conditional equality over negation. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- CondEq ( x = y -> ( ph <-> ps ) )   =>    |- CondEq ( x = y -> ( -. ph <-> -. ps ) )
Theorem	cdeqal 3424*	Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- CondEq ( x = y -> ( ph <-> ps ) )   =>    |- CondEq ( x = y -> ( A. z ph <-> A. z ps ) )
Theorem	cdeqab 3425*	Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- CondEq ( x = y -> ( ph <-> ps ) )   =>    |- CondEq ( x = y -> { z | ph } = { z | ps } )
Theorem	cdeqal1 3426*	Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- CondEq ( x = y -> ( ph <-> ps ) )   =>    |- CondEq ( x = y -> ( A. x ph <-> A. y ps ) )
Theorem	cdeqab1 3427*	Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- CondEq ( x = y -> ( ph <-> ps ) )   =>    |- CondEq ( x = y -> { x | ph } = { y | ps } )
Theorem	cdeqim 3428	Distribute conditional equality over implication. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- CondEq ( x = y -> ( ph <-> ps ) )   &    |- CondEq ( x = y -> ( ch <-> th ) )   =>    |- CondEq ( x = y -> ( ( ph -> ch ) <-> ( ps -> th ) ) )
Theorem	cdeqcv 3429	Conditional equality for set-to-class promotion. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- CondEq ( x = y -> x = y )
Theorem	cdeqeq 3430	Distribute conditional equality over equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- CondEq ( x = y -> A = B )   &    |- CondEq ( x = y -> C = D )   =>    |- CondEq ( x = y -> ( A = C <-> B = D ) )
Theorem	cdeqel 3431	Distribute conditional equality over elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- CondEq ( x = y -> A = B )   &    |- CondEq ( x = y -> C = D )   =>    |- CondEq ( x = y -> ( A e. C <-> B e. D ) )
Theorem	nfcdeq 3432*	If we have a conditional equality proof, where ph is ph ( x ) and ps is ph ( y ), and ph ( x ) in fact does not have x free in it according to F/, then ph ( x ) <-> ph ( y ) unconditionally. This proves that F/ x ph is actually a not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/ x ph   &    |- CondEq ( x = y -> ( ph <-> ps ) )   =>    |- ( ph <-> ps )
Theorem	nfccdeq 3433*	Variation of nfcdeq 3432 for classes. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/_ x A   &    |- CondEq ( x = y -> A = B )   =>    |- A = B
2.1.8  Russell's Paradox
Theorem	ru 3434	Russell's Paradox. Proposition 4.14 of [TakeutiZaring] p. 14. In the late 1800s, Frege's Axiom of (unrestricted) Comprehension, expressed in our notation as A e. _V, asserted that any collection of sets A is a set i.e. belongs to the universe _V of all sets. In particular, by substituting { x | x e/ x } (the "Russell class") for A, it asserted { x | x e/ x } e. _V, meaning that the "collection of all sets which are not members of themselves" is a set. However, here we prove { x | x e/ x } e/ _V. This contradiction was discovered by Russell in 1901 (published in 1903), invalidating the Comprehension Axiom and leading to the collapse of Frege's system. In 1908, Zermelo rectified this fatal flaw by replacing Comprehension with a weaker Subset (or Separation) Axiom ssex 4802 asserting that A is a set only when it is smaller than some other set B. However, Zermelo was then faced with a "chicken and egg" problem of how to show B is a set, leading him to introduce the set-building axioms of Null Set 0ex 4790, Pairing prex 4909, Union uniex 6953, Power Set pwex 4848, and Infinity omex 8540 to give him some starting sets to work with (all of which, before Russell's Paradox, were immediate consequences of Frege's Comprehension). In 1922 Fraenkel strengthened the Subset Axiom with our present Replacement Axiom funimaex 5976 (whose modern formalization is due to Skolem, also in 1922). Thus, in a very real sense Russell's Paradox spawned the invention of ZF set theory and completely revised the foundations of mathematics! Another mainstream formalization of set theory, devised by von Neumann, Bernays, and Goedel, uses class variables rather than setvar variables as its primitives. The axiom system NBG in [Mendelson] p. 225 is suitable for a Metamath encoding. NBG is a conservative extension of ZF in that it proves exactly the same theorems as ZF that are expressible in the language of ZF. An advantage of NBG is that it is finitely axiomatizable - the Axiom of Replacement can be broken down into a finite set of formulas that eliminate its wff metavariable. Finite axiomatizability is required by some proof languages (although not by Metamath). There is a stronger version of NBG called Morse-Kelley (axiom system MK in [Mendelson] p. 287). Russell himself continued in a different direction, avoiding the paradox with his "theory of types." Quine extended Russell's ideas to formulate his New Foundations set theory (axiom system NF of [Quine] p. 331). In NF, the collection of all sets is a set, contradicting ZF and NBG set theories, and it has other bizarre consequences: when sets become too huge (beyond the size of those used in standard mathematics), the Axiom of Choice ac4 9297 and Cantor's Theorem canth 6608 are provably false! (See ncanth 6609 for some intuition behind the latter.) Recent results (as of 2014) seem to show that NF is equiconsistent to Z (ZF in which ax-sep 4781 replaces ax-rep 4771) with ax-sep 4781 restricted to only bounded quantifiers. NF is finitely axiomatizable and can be encoded in Metamath using the axioms from T. Hailperin, "A set of axioms for logic," J. Symb. Logic 9:1-19 (1944). Under our ZF set theory, every set is a member of the Russell class by elirrv 8504 (derived from the Axiom of Regularity), so for us the Russell class equals the universe _V (theorem ruv 8507). See ruALT 8508 for an alternate proof of ru 3434 derived from that fact. (Contributed by NM, 7-Aug-1994.)
|- { x | x e/ x } e/ _V
2.1.9  Proper substitution of classes for sets
Syntax	wsbc 3435	Extend wff notation to include the proper substitution of a class for a set. Read this notation as "the proper substitution of class A for setvar variable x in wff ph."
wff [. A / x ]. ph
Definition	df-sbc 3436	Define the proper substitution of a class for a set. When A is a proper class, our definition evaluates to false. This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3462 for our definition, which always evaluates to true for proper classes. Our definition also does not produce the same results as discussed in the proof of Theorem 6.6 of [Quine] p. 42 (although Theorem 6.6 itself does hold, as shown by dfsbcq 3437 below). For example, if A is a proper class, Quine's substitution of A for y in 0 e. y evaluates to 0 e. A rather than our falsehood. (This can be seen by substituting A, y, and 0 for alpha, beta, and gamma in Subcase 1 of Quine's discussion on p. 42.) Unfortunately, Quine's definition requires a recursive syntactic breakdown of ph, and it does not seem possible to express it with a single closed formula. If we did not want to commit to any specific proper class behavior, we could use this definition only to prove theorem dfsbcq 3437, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3436 in the form of sbc8g 3443. However, the behavior of Quine's definition at proper classes is similarly arbitrary, and for practical reasons (to avoid having to prove sethood of A in every use of this definition) we allow direct reference to df-sbc 3436 and assert that [. A / x ]. ph is always false when A is a proper class. The theorem sbc2or 3444 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3437. The related definition df-csb 3534 defines proper substitution into a class variable (as opposed to a wff variable). (Contributed by NM, 14-Apr-1995.) (Revised by NM, 25-Dec-2016.)
|- ( [. A / x ]. ph <-> A e. { x | ph } )
Theorem	dfsbcq 3437	Proper substitution of a class for a set in a wff given equal classes. This is the essence of the sixth axiom of Frege, specifically Proposition 52 of [Frege1879] p. 50. This theorem, which is similar to Theorem 6.7 of [Quine] p. 42 and holds under both our definition and Quine's, provides us with a weak definition of the proper substitution of a class for a set. Since our df-sbc 3436 does not result in the same behavior as Quine's for proper classes, if we wished to avoid conflict with Quine's definition we could start with this theorem and dfsbcq2 3438 instead of df-sbc 3436. (dfsbcq2 3438 is needed because unlike Quine we do not overload the df-sb 1881 syntax.) As a consequence of these theorems, we can derive sbc8g 3443, which is a weaker version of df-sbc 3436 that leaves substitution undefined when A is a proper class. However, it is often a nuisance to have to prove the sethood hypothesis of sbc8g 3443, so we will allow direct use of df-sbc 3436 after theorem sbc2or 3444 below. Proper substitution with a proper class is rarely needed, and when it is, we can simply use the expansion of Quine's definition. (Contributed by NM, 14-Apr-1995.)
|- ( A = B -> ( [. A / x ]. ph <-> [. B / x ]. ph ) )
Theorem	dfsbcq2 3438	This theorem, which is similar to Theorem 6.7 of [Quine] p. 42 and holds under both our definition and Quine's, relates logic substitution df-sb 1881 and substitution for class variables df-sbc 3436. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 3437. (Contributed by NM, 31-Dec-2016.)
|- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
Theorem	sbsbc 3439	Show that df-sb 1881 and df-sbc 3436 are equivalent when the class term A in df-sbc 3436 is a setvar variable. This theorem lets us reuse theorems based on df-sb 1881 for proofs involving df-sbc 3436. (Contributed by NM, 31-Dec-2016.) (Proof modification is discouraged.)
|- ( [ y / x ] ph <-> [. y / x ]. ph )
Theorem	sbceq1d 3440	Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.)
|- ( ph -> A = B )   =>    |- ( ph -> ( [. A / x ]. ps <-> [. B / x ]. ps ) )
Theorem	sbceq1dd 3441	Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.)
|- ( ph -> A = B )   &    |- ( ph -> [. A / x ]. ps )   =>    |- ( ph -> [. B / x ]. ps )
Theorem	sbceqbid 3442*	Equality theorem for class substitution. (Contributed by Thierry Arnoux, 4-Sep-2018.)
|- ( ph -> A = B )   &    |- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( [. A / x ]. ps <-> [. B / x ]. ch ) )
Theorem	sbc8g 3443	This is the closest we can get to df-sbc 3436 if we start from dfsbcq 3437 (see its comments) and dfsbcq2 3438. (Contributed by NM, 18-Nov-2008.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.)
|- ( A e. V -> ( [. A / x ]. ph <-> A e. { x | ph } ) )
Theorem	sbc2or 3444*	The disjunction of two equivalences for class substitution does not require a class existence hypothesis. This theorem tells us that there are only 2 possibilities for [ A / x ] ph behavior at proper classes, matching the sbc5 3460 (false) and sbc6 3462 (true) conclusions. This is interesting since dfsbcq 3437 and dfsbcq2 3438 (from which it is derived) do not appear to say anything obvious about proper class behavior. Note that this theorem does not tell us that it is always one or the other at proper classes; it could "flip" between false (the first disjunct) and true (the second disjunct) as a function of some other variable y that ph or A may contain. (Contributed by NM, 11-Oct-2004.) (Proof modification is discouraged.)
|- ( ( [. A / x ]. ph <-> E. x ( x = A /\ ph ) ) \/ ( [. A / x ]. ph <-> A. x ( x = A -> ph ) ) )
Theorem	sbcex 3445	By our definition of proper substitution, it can only be true if the substituted expression is a set. (Contributed by Mario Carneiro, 13-Oct-2016.)
|- ( [. A / x ]. ph -> A e. _V )
Theorem	sbceq1a 3446	Equality theorem for class substitution. Class version of sbequ12 2111. (Contributed by NM, 26-Sep-2003.)
|- ( x = A -> ( ph <-> [. A / x ]. ph ) )
Theorem	sbceq2a 3447	Equality theorem for class substitution. Class version of sbequ12r 2112. (Contributed by NM, 4-Jan-2017.)
|- ( A = x -> ( [. A / x ]. ph <-> ph ) )
Theorem	spsbc 3448	Specialization: if a formula is true for all sets, it is true for any class which is a set. Similar to Theorem 6.11 of [Quine] p. 44. This is Frege's ninth axiom per Proposition 58 of [Frege1879] p. 51. See also stdpc4 2353 and rspsbc 3518. (Contributed by NM, 16-Jan-2004.)
|- ( A e. V -> ( A. x ph -> [. A / x ]. ph ) )
Theorem	spsbcd 3449	Specialization: if a formula is true for all sets, it is true for any class which is a set. Similar to Theorem 6.11 of [Quine] p. 44. See also stdpc4 2353 and rspsbc 3518. (Contributed by Mario Carneiro, 9-Feb-2017.)
|- ( ph -> A e. V )   &    |- ( ph -> A. x ps )   =>    |- ( ph -> [. A / x ]. ps )
Theorem	sbcth 3450	A substitution into a theorem remains true (when A is a set). (Contributed by NM, 5-Nov-2005.)
|- ph   =>    |- ( A e. V -> [. A / x ]. ph )
Theorem	sbcthdv 3451*	Deduction version of sbcth 3450. (Contributed by NM, 30-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
|- ( ph -> ps )   =>    |- ( ( ph /\ A e. V ) -> [. A / x ]. ps )
Theorem	sbcid 3452	An identity theorem for substitution. See sbid 2114. (Contributed by Mario Carneiro, 18-Feb-2017.)
|- ( [. x / x ]. ph <-> ph )
Theorem	nfsbc1d 3453	Deduction version of nfsbc1 3454. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 12-Oct-2016.)
|- ( ph -> F/_ x A )   =>    |- ( ph -> F/ x [. A / x ]. ps )
Theorem	nfsbc1 3454	Bound-variable hypothesis builder for class substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.)
|- F/_ x A   =>    |- F/ x [. A / x ]. ph
Theorem	nfsbc1v 3455*	Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.)
|- F/ x [. A / x ]. ph
Theorem	nfsbcd 3456	Deduction version of nfsbc 3457. (Contributed by NM, 23-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.)
|- F/ y ph   &    |- ( ph -> F/_ x A )   &    |- ( ph -> F/ x ps )   =>    |- ( ph -> F/ x [. A / y ]. ps )
Theorem	nfsbc 3457	Bound-variable hypothesis builder for class substitution. (Contributed by NM, 7-Sep-2014.) (Revised by Mario Carneiro, 12-Oct-2016.)
|- F/_ x A   &    |- F/ x ph   =>    |- F/ x [. A / y ]. ph
Theorem	sbcco 3458*	A composition law for class substitution. (Contributed by NM, 26-Sep-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)
|- ( [. A / y ]. [. y / x ]. ph <-> [. A / x ]. ph )
Theorem	sbcco2 3459*	A composition law for class substitution. Importantly, x may occur free in the class expression substituted for A. (Contributed by NM, 5-Sep-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
|- ( x = y -> A = B )   =>    |- ( [. x / y ]. [. B / x ]. ph <-> [. A / x ]. ph )
Theorem	sbc5 3460*	An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.)
|- ( [. A / x ]. ph <-> E. x ( x = A /\ ph ) )
Theorem	sbc6g 3461*	An equivalence for class substitution. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
|- ( A e. V -> ( [. A / x ]. ph <-> A. x ( x = A -> ph ) ) )
Theorem	sbc6 3462*	An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Proof shortened by Eric Schmidt, 17-Jan-2007.)
|- A e. _V   =>    |- ( [. A / x ]. ph <-> A. x ( x = A -> ph ) )
Theorem	sbc7 3463*	An equivalence for class substitution in the spirit of df-clab 2609. Note that x and A don't have to be distinct. (Contributed by NM, 18-Nov-2008.) (Revised by Mario Carneiro, 13-Oct-2016.)
|- ( [. A / x ]. ph <-> E. y ( y = A /\ [. y / x ]. ph ) )
Theorem	cbvsbc 3464	Change bound variables in a wff substitution. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
|- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( [. A / x ]. ph <-> [. A / y ]. ps )
Theorem	cbvsbcv 3465*	Change the bound variable of a class substitution using implicit substitution. (Contributed by NM, 30-Sep-2008.) (Revised by Mario Carneiro, 13-Oct-2016.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( [. A / x ]. ph <-> [. A / y ]. ps )
Theorem	sbciegft 3466*	Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 3467.) (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
|- ( ( A e. V /\ F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) ) -> ( [. A / x ]. ph <-> ps ) )
Theorem	sbciegf 3467*	Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
|- F/ x ps   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. V -> ( [. A / x ]. ph <-> ps ) )
Theorem	sbcieg 3468*	Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 10-Nov-2005.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. V -> ( [. A / x ]. ph <-> ps ) )
Theorem	sbcie2g 3469*	Conversion of implicit substitution to explicit class substitution. This version of sbcie 3470 avoids a disjointness condition on x , A by substituting twice. (Contributed by Mario Carneiro, 15-Oct-2016.)
|- ( x = y -> ( ph <-> ps ) )   &    |- ( y = A -> ( ps <-> ch ) )   =>    |- ( A e. V -> ( [. A / x ]. ph <-> ch ) )
Theorem	sbcie 3470*	Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 4-Sep-2004.)
|- A e. _V   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( [. A / x ]. ph <-> ps )
Theorem	sbciedf 3471*	Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 29-Dec-2014.)
|- ( ph -> A e. V )   &    |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )   &    |- F/ x ph   &    |- ( ph -> F/ x ch )   =>    |- ( ph -> ( [. A / x ]. ps <-> ch ) )
Theorem	sbcied 3472*	Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.)
|- ( ph -> A e. V )   &    |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( [. A / x ]. ps <-> ch ) )
Theorem	sbcied2 3473*	Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.)
|- ( ph -> A e. V )   &    |- ( ph -> A = B )   &    |- ( ( ph /\ x = B ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( [. A / x ]. ps <-> ch ) )
Theorem	elrabsf 3474	Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 3360 has implicit substitution). The hypothesis specifies that x must not be a free variable in B. (Contributed by NM, 30-Sep-2003.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
|- F/_ x B   =>    |- ( A e. { x e. B | ph } <-> ( A e. B /\ [. A / x ]. ph ) )
Theorem	eqsbc3 3475*	Substitution applied to an atomic wff. Set theory version of eqsb3 2728. (Contributed by Andrew Salmon, 29-Jun-2011.)
|- ( A e. V -> ( [. A / x ]. x = B <-> A = B ) )
Theorem	sbcng 3476	Move negation in and out of class substitution. (Contributed by NM, 16-Jan-2004.)
|- ( A e. V -> ( [. A / x ]. -. ph <-> -. [. A / x ]. ph ) )
Theorem	sbcimg 3477	Distribution of class substitution over implication. (Contributed by NM, 16-Jan-2004.)
|- ( A e. V -> ( [. A / x ]. ( ph -> ps ) <-> ( [. A / x ]. ph -> [. A / x ]. ps ) ) )
Theorem	sbcan 3478	Distribution of class substitution over conjunction. (Contributed by NM, 31-Dec-2016.) (Revised by NM, 17-Aug-2018.)
|- ( [. A / x ]. ( ph /\ ps ) <-> ( [. A / x ]. ph /\ [. A / x ]. ps ) )
Theorem	sbcor 3479	Distribution of class substitution over disjunction. (Contributed by NM, 31-Dec-2016.) (Revised by NM, 17-Aug-2018.)
|- ( [. A / x ]. ( ph \/ ps ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps ) )
Theorem	sbcbig 3480	Distribution of class substitution over biconditional. (Contributed by Raph Levien, 10-Apr-2004.)
|- ( A e. V -> ( [. A / x ]. ( ph <-> ps ) <-> ( [. A / x ]. ph <-> [. A / x ]. ps ) ) )
Theorem	sbcn1 3481	Move negation in and out of class substitution. One direction of sbcng 3476 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)
|- ( [. A / x ]. -. ph -> -. [. A / x ]. ph )
Theorem	sbcim1 3482	Distribution of class substitution over implication. One direction of sbcimg 3477 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)
|- ( [. A / x ]. ( ph -> ps ) -> ( [. A / x ]. ph -> [. A / x ]. ps ) )
Theorem	sbcbi1 3483	Distribution of class substitution over biconditional. One direction of sbcbig 3480 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)
|- ( [. A / x ]. ( ph <-> ps ) -> ( [. A / x ]. ph <-> [. A / x ]. ps ) )
Theorem	sbcbi2 3484	Substituting into equivalent wff's gives equivalent results. (Contributed by Giovanni Mascellani, 9-Apr-2018.)
|- ( A. x ( ph <-> ps ) -> ( [. A / x ]. ph <-> [. A / x ]. ps ) )
Theorem	sbcal 3485*	Move universal quantifier in and out of class substitution. (Contributed by NM, 31-Dec-2016.) (Revised by NM, 18-Aug-2018.)
|- ( [. A / y ]. A. x ph <-> A. x [. A / y ]. ph )
Theorem	sbcex2 3486*	Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.) (Revised by NM, 18-Aug-2018.)
|- ( [. A / y ]. E. x ph <-> E. x [. A / y ]. ph )
Theorem	sbceqal 3487*	Set theory version of sbeqal1 38598. (Contributed by Andrew Salmon, 28-Jun-2011.)
|- ( A e. V -> ( A. x ( x = A -> x = B ) -> A = B ) )
Theorem	sbeqalb 3488*	Theorem *14.121 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 28-Jun-2011.) (Proof shortened by Wolf Lammen, 9-May-2013.)
|- ( A e. V -> ( ( A. x ( ph <-> x = A ) /\ A. x ( ph <-> x = B ) ) -> A = B ) )
Theorem	sbcbid 3489	Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.)
|- F/ x ph   &    |- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( [. A / x ]. ps <-> [. A / x ]. ch ) )
Theorem	sbcbidv 3490*	Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( [. A / x ]. ps <-> [. A / x ]. ch ) )
Theorem	sbcbii 3491	Formula-building inference rule for class substitution. (Contributed by NM, 11-Nov-2005.)
|- ( ph <-> ps )   =>    |- ( [. A / x ]. ph <-> [. A / x ]. ps )
Theorem	eqsbc3r 3492*	eqsbc3 3475 with setvar variable on right side of equals sign. (Contributed by Alan Sare, 24-Oct-2011.) (Proof shortened by JJ, 7-Jul-2021.)
|- ( A e. V -> ( [. A / x ]. B = x <-> B = A ) )
Theorem	eqsbc3rOLD 3493*	Obsolete proof of eqsbc3r 3492 as of 7-Jul-2021. This proof was automatically generated from the virtual deduction proof eqsbc3rVD 39075 using a translation program. (Contributed by Alan Sare, 24-Oct-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
|- ( A e. V -> ( [. A / x ]. B = x <-> B = A ) )
Theorem	sbc3an 3494	Distribution of class substitution over triple conjunction. (Contributed by NM, 14-Dec-2006.) (Revised by NM, 17-Aug-2018.)
|- ( [. A / x ]. ( ph /\ ps /\ ch ) <-> ( [. A / x ]. ph /\ [. A / x ]. ps /\ [. A / x ]. ch ) )
Theorem	sbcel1v 3495*	Class substitution into a membership relation. (Contributed by NM, 17-Aug-2018.)
|- ( [. A / x ]. x e. B <-> A e. B )
Theorem	sbcel2gv 3496*	Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- ( B e. V -> ( [. B / x ]. A e. x <-> A e. B ) )
Theorem	sbcel21v 3497*	Class substitution into a membership relation. One direction of sbcel2gv 3496 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)
|- ( [. B / x ]. A e. x -> A e. B )
Theorem	sbcimdv 3498*	Substitution analogue of Theorem 19.20 of [Margaris] p. 90 (alim 1738). (Contributed by NM, 11-Nov-2005.) (Revised by NM, 17-Aug-2018.) (Proof shortened by JJ, 7-Jul-2021.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) )
Theorem	sbcimdvOLD 3499*	Obsolete proof of sbcimdv 3498 as of 7-Jul-2021. (Contributed by NM, 11-Nov-2005.) (Revised by NM, 17-Aug-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) )
Theorem	sbctt 3500	Substitution for a variable not free in a wff does not affect it. (Contributed by Mario Carneiro, 14-Oct-2016.)
|- ( ( A e. V /\ F/ x ph ) -> ( [. A / x ]. ph <-> ph ) )