P. 45 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 4401-4500   *Has distinct variable group(s)

Type	Label	Description
Statement
Definition	df-iord 4401*	Define the ordinal predicate, which is true for a class that is transitive and whose elements are transitive. Definition of ordinal in [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic". Some sources will define a notation for ordinal order corresponding to < and <_ but we just use e. and C_ respectively. (Contributed by Jim Kingdon, 10-Oct-2018.) Use its alias dford3 4402 instead for naming consistency with set.mm. (New usage is discouraged.)
|- ( Ord A <-> ( Tr A /\ A. x e. A Tr x ) )
Theorem	dford3 4402*	Alias for df-iord 4401. Use it instead of df-iord 4401 for naming consistency with set.mm. (Contributed by Jim Kingdon, 10-Oct-2018.)
|- ( Ord A <-> ( Tr A /\ A. x e. A Tr x ) )
Definition	df-on 4403	Define the class of all ordinal numbers. Definition 7.11 of [TakeutiZaring] p. 38. (Contributed by NM, 5-Jun-1994.)
|- On = { x | Ord x }
Definition	df-ilim 4404	Define the limit ordinal predicate, which is true for an ordinal that has the empty set as an element and is not a successor (i.e. that is the union of itself). Our definition combines the definition of Lim of [BellMachover] p. 471 and Exercise 1 of [TakeutiZaring] p. 42, and then changes A =/= (/) to (/) e. A (which would be equivalent given the law of the excluded middle, but which is not for us). (Contributed by Jim Kingdon, 11-Nov-2018.) Use its alias dflim2 4405 instead for naming consistency with set.mm. (New usage is discouraged.)
|- ( Lim A <-> ( Ord A /\ (/) e. A /\ A = U. A ) )
Theorem	dflim2 4405	Alias for df-ilim 4404. Use it instead of df-ilim 4404 for naming consistency with set.mm. (Contributed by NM, 4-Nov-2004.)
|- ( Lim A <-> ( Ord A /\ (/) e. A /\ A = U. A ) )
Definition	df-suc 4406	Define the successor of a class. When applied to an ordinal number, the successor means the same thing as "plus 1". Definition 7.22 of [TakeutiZaring] p. 41, who use "+ 1" to denote this function. Our definition is a generalization to classes. Although it is not conventional to use it with proper classes, it has no effect on a proper class (sucprc 4447). Some authors denote the successor operation with a prime (apostrophe-like) symbol, such as Definition 6 of [Suppes] p. 134 and the definition of successor in [Mendelson] p. 246 (who uses the symbol "Suc" as a predicate to mean "is a successor ordinal"). The definition of successor of [Enderton] p. 68 denotes the operation with a plus-sign superscript. (Contributed by NM, 30-Aug-1993.)
|- suc A = ( A u. { A } )
Theorem	ordeq 4407	Equality theorem for the ordinal predicate. (Contributed by NM, 17-Sep-1993.)
|- ( A = B -> ( Ord A <-> Ord B ) )
Theorem	elong 4408	An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)
|- ( A e. V -> ( A e. On <-> Ord A ) )
Theorem	elon 4409	An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)
|- A e. _V   =>    |- ( A e. On <-> Ord A )
Theorem	eloni 4410	An ordinal number has the ordinal property. (Contributed by NM, 5-Jun-1994.)
|- ( A e. On -> Ord A )
Theorem	elon2 4411	An ordinal number is an ordinal set. (Contributed by NM, 8-Feb-2004.)
|- ( A e. On <-> ( Ord A /\ A e. _V ) )
Theorem	limeq 4412	Equality theorem for the limit predicate. (Contributed by NM, 22-Apr-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|- ( A = B -> ( Lim A <-> Lim B ) )
Theorem	ordtr 4413	An ordinal class is transitive. (Contributed by NM, 3-Apr-1994.)
|- ( Ord A -> Tr A )
Theorem	ordelss 4414	An element of an ordinal class is a subset of it. (Contributed by NM, 30-May-1994.)
|- ( ( Ord A /\ B e. A ) -> B C_ A )
Theorem	trssord 4415	A transitive subclass of an ordinal class is ordinal. (Contributed by NM, 29-May-1994.)
|- ( ( Tr A /\ A C_ B /\ Ord B ) -> Ord A )
Theorem	ordelord 4416	An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. (Contributed by NM, 23-Apr-1994.)
|- ( ( Ord A /\ B e. A ) -> Ord B )
Theorem	tron 4417	The class of all ordinal numbers is transitive. (Contributed by NM, 4-May-2009.)
|- Tr On
Theorem	ordelon 4418	An element of an ordinal class is an ordinal number. (Contributed by NM, 26-Oct-2003.)
|- ( ( Ord A /\ B e. A ) -> B e. On )
Theorem	onelon 4419	An element of an ordinal number is an ordinal number. Theorem 2.2(iii) of [BellMachover] p. 469. (Contributed by NM, 26-Oct-2003.)
|- ( ( A e. On /\ B e. A ) -> B e. On )
Theorem	ordin 4420	The intersection of two ordinal classes is ordinal. Proposition 7.9 of [TakeutiZaring] p. 37. (Contributed by NM, 9-May-1994.)
|- ( ( Ord A /\ Ord B ) -> Ord ( A i^i B ) )
Theorem	onin 4421	The intersection of two ordinal numbers is an ordinal number. (Contributed by NM, 7-Apr-1995.)
|- ( ( A e. On /\ B e. On ) -> ( A i^i B ) e. On )
Theorem	onelss 4422	An element of an ordinal number is a subset of the number. (Contributed by NM, 5-Jun-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|- ( A e. On -> ( B e. A -> B C_ A ) )
Theorem	ordtr1 4423	Transitive law for ordinal classes. (Contributed by NM, 12-Dec-2004.)
|- ( Ord C -> ( ( A e. B /\ B e. C ) -> A e. C ) )
Theorem	ontr1 4424	Transitive law for ordinal numbers. Theorem 7M(b) of [Enderton] p. 192. (Contributed by NM, 11-Aug-1994.)
|- ( C e. On -> ( ( A e. B /\ B e. C ) -> A e. C ) )
Theorem	onintss 4425*	If a property is true for an ordinal number, then the minimum ordinal number for which it is true is smaller or equal. Theorem Schema 61 of [Suppes] p. 228. (Contributed by NM, 3-Oct-2003.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. On -> ( ps -> |^| { x e. On | ph } C_ A ) )
Theorem	ord0 4426	The empty set is an ordinal class. (Contributed by NM, 11-May-1994.)
|- Ord (/)
Theorem	0elon 4427	The empty set is an ordinal number. Corollary 7N(b) of [Enderton] p. 193. (Contributed by NM, 17-Sep-1993.)
|- (/) e. On
Theorem	inton 4428	The intersection of the class of ordinal numbers is the empty set. (Contributed by NM, 20-Oct-2003.)
|- |^| On = (/)
Theorem	nlim0 4429	The empty set is not a limit ordinal. (Contributed by NM, 24-Mar-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|- -. Lim (/)
Theorem	limord 4430	A limit ordinal is ordinal. (Contributed by NM, 4-May-1995.)
|- ( Lim A -> Ord A )
Theorem	limuni 4431	A limit ordinal is its own supremum (union). (Contributed by NM, 4-May-1995.)
|- ( Lim A -> A = U. A )
Theorem	limuni2 4432	The union of a limit ordinal is a limit ordinal. (Contributed by NM, 19-Sep-2006.)
|- ( Lim A -> Lim U. A )
Theorem	0ellim 4433	A limit ordinal contains the empty set. (Contributed by NM, 15-May-1994.)
|- ( Lim A -> (/) e. A )
Theorem	limelon 4434	A limit ordinal class that is also a set is an ordinal number. (Contributed by NM, 26-Apr-2004.)
|- ( ( A e. B /\ Lim A ) -> A e. On )
Theorem	onn0 4435	The class of all ordinal numbers is not empty. (Contributed by NM, 17-Sep-1995.)
|- On =/= (/)
Theorem	onm 4436	The class of all ordinal numbers is inhabited. (Contributed by Jim Kingdon, 6-Mar-2019.)
|- E. x x e. On
Theorem	suceq 4437	Equality of successors. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|- ( A = B -> suc A = suc B )
Theorem	elsuci 4438	Membership in a successor. This one-way implication does not require that either A or B be sets. (Contributed by NM, 6-Jun-1994.)
|- ( A e. suc B -> ( A e. B \/ A = B ) )
Theorem	elsucg 4439	Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-1995.)
|- ( A e. V -> ( A e. suc B <-> ( A e. B \/ A = B ) ) )
Theorem	elsuc2g 4440	Variant of membership in a successor, requiring that B rather than A be a set. (Contributed by NM, 28-Oct-2003.)
|- ( B e. V -> ( A e. suc B <-> ( A e. B \/ A = B ) ) )
Theorem	elsuc 4441	Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-2003.)
|- A e. _V   =>    |- ( A e. suc B <-> ( A e. B \/ A = B ) )
Theorem	elsuc2 4442	Membership in a successor. (Contributed by NM, 15-Sep-2003.)
|- A e. _V   =>    |- ( B e. suc A <-> ( B e. A \/ B = A ) )
Theorem	nfsuc 4443	Bound-variable hypothesis builder for successor. (Contributed by NM, 15-Sep-2003.)
|- F/_ x A   =>    |- F/_ x suc A
Theorem	elelsuc 4444	Membership in a successor. (Contributed by NM, 20-Jun-1998.)
|- ( A e. B -> A e. suc B )
Theorem	sucel 4445*	Membership of a successor in another class. (Contributed by NM, 29-Jun-2004.)
|- ( suc A e. B <-> E. x e. B A. y ( y e. x <-> ( y e. A \/ y = A ) ) )
Theorem	suc0 4446	The successor of the empty set. (Contributed by NM, 1-Feb-2005.)
|- suc (/) = { (/) }
Theorem	sucprc 4447	A proper class is its own successor. (Contributed by NM, 3-Apr-1995.)
|- ( -. A e. _V -> suc A = A )
Theorem	unisuc 4448	A transitive class is equal to the union of its successor. Combines Theorem 4E of [Enderton] p. 72 and Exercise 6 of [Enderton] p. 73. (Contributed by NM, 30-Aug-1993.)
|- A e. _V   =>    |- ( Tr A <-> U. suc A = A )
Theorem	unisucg 4449	A transitive class is equal to the union of its successor. Combines Theorem 4E of [Enderton] p. 72 and Exercise 6 of [Enderton] p. 73. (Contributed by Jim Kingdon, 18-Aug-2019.)
|- ( A e. V -> ( Tr A <-> U. suc A = A ) )
Theorem	sssucid 4450	A class is included in its own successor. Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized to arbitrary classes). (Contributed by NM, 31-May-1994.)
|- A C_ suc A
Theorem	sucidg 4451	Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized). (Contributed by NM, 25-Mar-1995.) (Proof shortened by Scott Fenton, 20-Feb-2012.)
|- ( A e. V -> A e. suc A )
Theorem	sucid 4452	A set belongs to its successor. (Contributed by NM, 22-Jun-1994.) (Proof shortened by Alan Sare, 18-Feb-2012.) (Proof shortened by Scott Fenton, 20-Feb-2012.)
|- A e. _V   =>    |- A e. suc A
Theorem	nsuceq0g 4453	No successor is empty. (Contributed by Jim Kingdon, 14-Oct-2018.)
|- ( A e. V -> suc A =/= (/) )
Theorem	eqelsuc 4454	A set belongs to the successor of an equal set. (Contributed by NM, 18-Aug-1994.)
|- A e. _V   =>    |- ( A = B -> A e. suc B )
Theorem	iunsuc 4455*	Inductive definition for the indexed union at a successor. (Contributed by Mario Carneiro, 4-Feb-2013.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
|- A e. _V   &    |- ( x = A -> B = C )   =>    |- U_ x e. suc A B = ( U_ x e. A B u. C )
Theorem	suctr 4456	The successor of a transitive class is transitive. (Contributed by Alan Sare, 11-Apr-2009.)
|- ( Tr A -> Tr suc A )
Theorem	trsuc 4457	A set whose successor belongs to a transitive class also belongs. (Contributed by NM, 5-Sep-2003.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
|- ( ( Tr A /\ suc B e. A ) -> B e. A )
Theorem	trsucss 4458	A member of the successor of a transitive class is a subclass of it. (Contributed by NM, 4-Oct-2003.)
|- ( Tr A -> ( B e. suc A -> B C_ A ) )
Theorem	sucssel 4459	A set whose successor is a subset of another class is a member of that class. (Contributed by NM, 16-Sep-1995.)
|- ( A e. V -> ( suc A C_ B -> A e. B ) )
Theorem	orduniss 4460	An ordinal class includes its union. (Contributed by NM, 13-Sep-2003.)
|- ( Ord A -> U. A C_ A )
Theorem	onordi 4461	An ordinal number is an ordinal class. (Contributed by NM, 11-Jun-1994.)
|- A e. On   =>    |- Ord A
Theorem	ontrci 4462	An ordinal number is a transitive class. (Contributed by NM, 11-Jun-1994.)
|- A e. On   =>    |- Tr A
Theorem	oneli 4463	A member of an ordinal number is an ordinal number. Theorem 7M(a) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.)
|- A e. On   =>    |- ( B e. A -> B e. On )
Theorem	onelssi 4464	A member of an ordinal number is a subset of it. (Contributed by NM, 11-Aug-1994.)
|- A e. On   =>    |- ( B e. A -> B C_ A )
Theorem	onelini 4465	An element of an ordinal number equals the intersection with it. (Contributed by NM, 11-Jun-1994.)
|- A e. On   =>    |- ( B e. A -> B = ( B i^i A ) )
Theorem	oneluni 4466	An ordinal number equals its union with any element. (Contributed by NM, 13-Jun-1994.)
|- A e. On   =>    |- ( B e. A -> ( A u. B ) = A )
Theorem	onunisuci 4467	An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994.)
|- A e. On   =>    |- U. suc A = A
2.4  IZF Set Theory - add the Axiom of Union
2.4.1  Introduce the Axiom of Union
Axiom	ax-un 4468*	Axiom of Union. An axiom of Intuitionistic Zermelo-Fraenkel set theory. It states that a set y exists that includes the union of a given set x i.e. the collection of all members of the members of x. The variant axun2 4470 states that the union itself exists. A version with the standard abbreviation for union is uniex2 4471. A version using class notation is uniex 4472. This is Axiom 3 of [Crosilla] p. "Axioms of CZF and IZF", except (a) unnecessary quantifiers are removed, (b) Crosilla has a biconditional rather than an implication (but the two are equivalent by bm1.3ii 4154), and (c) the order of the conjuncts is swapped (which is equivalent by ancom 266). The union of a class df-uni 3840 should not be confused with the union of two classes df-un 3161. Their relationship is shown in unipr 3853. (Contributed by NM, 23-Dec-1993.)
|- E. y A. z ( E. w ( z e. w /\ w e. x ) -> z e. y )
Theorem	zfun 4469*	Axiom of Union expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.)
|- E. x A. y ( E. x ( y e. x /\ x e. z ) -> y e. x )
Theorem	axun2 4470*	A variant of the Axiom of Union ax-un 4468. For any set x, there exists a set y whose members are exactly the members of the members of x i.e. the union of x. Axiom Union of [BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)
|- E. y A. z ( z e. y <-> E. w ( z e. w /\ w e. x ) )
Theorem	uniex2 4471*	The Axiom of Union using the standard abbreviation for union. Given any set x, its union y exists. (Contributed by NM, 4-Jun-2006.)
|- E. y y = U. x
Theorem	uniex 4472	The Axiom of Union in class notation. This says that if A is a set i.e. A e. _V (see isset 2769), then the union of A is also a set. Same as Axiom 3 of [TakeutiZaring] p. 16. (Contributed by NM, 11-Aug-1993.)
|- A e. _V   =>    |- U. A e. _V
Theorem	vuniex 4473	The union of a setvar is a set. (Contributed by BJ, 3-May-2021.)
|- U. x e. _V
Theorem	uniexg 4474	The ZF Axiom of Union in class notation, in the form of a theorem instead of an inference. We use the antecedent A e. V instead of A e. _V to make the theorem more general and thus shorten some proofs; obviously the universal class constant _V is one possible substitution for class variable V. (Contributed by NM, 25-Nov-1994.)
|- ( A e. V -> U. A e. _V )
Theorem	uniexd 4475	Deduction version of the ZF Axiom of Union in class notation. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
|- ( ph -> A e. V )   =>    |- ( ph -> U. A e. _V )
Theorem	unex 4476	The union of two sets is a set. Corollary 5.8 of [TakeutiZaring] p. 16. (Contributed by NM, 1-Jul-1994.)
|- A e. _V   &    |- B e. _V   =>    |- ( A u. B ) e. _V
Theorem	unexb 4477	Existence of union is equivalent to existence of its components. (Contributed by NM, 11-Jun-1998.)
|- ( ( A e. _V /\ B e. _V ) <-> ( A u. B ) e. _V )
Theorem	unexg 4478	A union of two sets is a set. Corollary 5.8 of [TakeutiZaring] p. 16. (Contributed by NM, 18-Sep-2006.)
|- ( ( A e. V /\ B e. W ) -> ( A u. B ) e. _V )
Theorem	tpexg 4479	An unordered triple of classes exists. (Contributed by NM, 10-Apr-1994.)
|- ( ( A e. U /\ B e. V /\ C e. W ) -> { A , B , C } e. _V )
Theorem	unisn3 4480*	Union of a singleton in the form of a restricted class abstraction. (Contributed by NM, 3-Jul-2008.)
|- ( A e. B -> U. { x e. B | x = A } = A )
Theorem	abnexg 4481*	Sufficient condition for a class abstraction to be a proper class. The class F can be thought of as an expression in x and the abstraction appearing in the statement as the class of values F as x varies through A. Assuming the antecedents, if that class is a set, then so is the "domain" A. The converse holds without antecedent, see abrexexg 6175. Note that the second antecedent A. x e. A x e. F cannot be translated to A C_ F since F may depend on x. In applications, one may take F = { x } or F = ~P x (see snnex 4483 and pwnex 4484 respectively, proved from abnex 4482, which is a consequence of abnexg 4481 with A = _V). (Contributed by BJ, 2-Dec-2021.)
|- ( A. x e. A ( F e. V /\ x e. F ) -> ( { y | E. x e. A y = F } e. W -> A e. _V ) )
Theorem	abnex 4482*	Sufficient condition for a class abstraction to be a proper class. Lemma for snnex 4483 and pwnex 4484. See the comment of abnexg 4481. (Contributed by BJ, 2-May-2021.)
|- ( A. x ( F e. V /\ x e. F ) -> -. { y | E. x y = F } e. _V )
Theorem	snnex 4483*	The class of all singletons is a proper class. (Contributed by NM, 10-Oct-2008.) (Proof shortened by Eric Schmidt, 7-Dec-2008.)
|- { x | E. y x = { y } } e/ _V
Theorem	pwnex 4484*	The class of all power sets is a proper class. See also snnex 4483. (Contributed by BJ, 2-May-2021.)
|- { x | E. y x = ~P y } e/ _V
Theorem	opeluu 4485	Each member of an ordered pair belongs to the union of the union of a class to which the ordered pair belongs. Lemma 3D of [Enderton] p. 41. (Contributed by NM, 31-Mar-1995.) (Revised by Mario Carneiro, 27-Feb-2016.)
|- A e. _V   &    |- B e. _V   =>    |- ( <. A , B >. e. C -> ( A e. U. U. C /\ B e. U. U. C ) )
Theorem	uniuni 4486*	Expression for double union that moves union into a class builder. (Contributed by FL, 28-May-2007.)
|- U. U. A = U. { x | E. y ( x = U. y /\ y e. A ) }
Theorem	eusv1 4487*	Two ways to express single-valuedness of a class expression A ( x ). (Contributed by NM, 14-Oct-2010.)
|- ( E! y A. x y = A <-> E. y A. x y = A )
Theorem	eusvnf 4488*	Even if x is free in A, it is effectively bound when A ( x ) is single-valued. (Contributed by NM, 14-Oct-2010.) (Revised by Mario Carneiro, 14-Oct-2016.)
|- ( E! y A. x y = A -> F/_ x A )
Theorem	eusvnfb 4489*	Two ways to say that A ( x ) is a set expression that does not depend on x. (Contributed by Mario Carneiro, 18-Nov-2016.)
|- ( E! y A. x y = A <-> ( F/_ x A /\ A e. _V ) )
Theorem	eusv2i 4490*	Two ways to express single-valuedness of a class expression A ( x ). (Contributed by NM, 14-Oct-2010.) (Revised by Mario Carneiro, 18-Nov-2016.)
|- ( E! y A. x y = A -> E! y E. x y = A )
Theorem	eusv2nf 4491*	Two ways to express single-valuedness of a class expression A ( x ). (Contributed by Mario Carneiro, 18-Nov-2016.)
|- A e. _V   =>    |- ( E! y E. x y = A <-> F/_ x A )
Theorem	eusv2 4492*	Two ways to express single-valuedness of a class expression A ( x ). (Contributed by NM, 15-Oct-2010.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
|- A e. _V   =>    |- ( E! y E. x y = A <-> E! y A. x y = A )
Theorem	reusv1 4493*	Two ways to express single-valuedness of a class expression C ( y ). (Contributed by NM, 16-Dec-2012.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
|- ( E. y e. B ph -> ( E! x e. A A. y e. B ( ph -> x = C ) <-> E. x e. A A. y e. B ( ph -> x = C ) ) )
Theorem	reusv3i 4494*	Two ways of expressing existential uniqueness via an indirect equality. (Contributed by NM, 23-Dec-2012.)
|- ( y = z -> ( ph <-> ps ) )   &    |- ( y = z -> C = D )   =>    |- ( E. x e. A A. y e. B ( ph -> x = C ) -> A. y e. B A. z e. B ( ( ph /\ ps ) -> C = D ) )
Theorem	reusv3 4495*	Two ways to express single-valuedness of a class expression C ( y ). See reusv1 4493 for the connection to uniqueness. (Contributed by NM, 27-Dec-2012.)
|- ( y = z -> ( ph <-> ps ) )   &    |- ( y = z -> C = D )   =>    |- ( E. y e. B ( ph /\ C e. A ) -> ( A. y e. B A. z e. B ( ( ph /\ ps ) -> C = D ) <-> E. x e. A A. y e. B ( ph -> x = C ) ) )
Theorem	alxfr 4496*	Transfer universal quantification from a variable x to another variable y contained in expression A. (Contributed by NM, 18-Feb-2007.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( ( A. y A e. B /\ A. x E. y x = A ) -> ( A. x ph <-> A. y ps ) )
Theorem	ralxfrd 4497*	Transfer universal quantification from a variable x to another variable y contained in expression A. (Contributed by NM, 15-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
|- ( ( ph /\ y e. C ) -> A e. B )   &    |- ( ( ph /\ x e. B ) -> E. y e. C x = A )   &    |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( A. x e. B ps <-> A. y e. C ch ) )
Theorem	rexxfrd 4498*	Transfer universal quantification from a variable x to another variable y contained in expression A. (Contributed by FL, 10-Apr-2007.) (Revised by Mario Carneiro, 15-Aug-2014.)
|- ( ( ph /\ y e. C ) -> A e. B )   &    |- ( ( ph /\ x e. B ) -> E. y e. C x = A )   &    |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( E. x e. B ps <-> E. y e. C ch ) )
Theorem	ralxfr2d 4499*	Transfer universal quantification from a variable x to another variable y contained in expression A. (Contributed by Mario Carneiro, 20-Aug-2014.)
|- ( ( ph /\ y e. C ) -> A e. V )   &    |- ( ph -> ( x e. B <-> E. y e. C x = A ) )   &    |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( A. x e. B ps <-> A. y e. C ch ) )
Theorem	rexxfr2d 4500*	Transfer universal quantification from a variable x to another variable y contained in expression A. (Contributed by Mario Carneiro, 20-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
|- ( ( ph /\ y e. C ) -> A e. V )   &    |- ( ph -> ( x e. B <-> E. y e. C x = A ) )   &    |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( E. x e. B ps <-> E. y e. C ch ) )