P. 42 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 4101-4200   *Has distinct variable group(s)

Type	Label	Description
Statement
Syntax	wlim 4101	Extend the definition of a wff to include the limit ordinal predicate.
wff Lim A
Syntax	csuc 4102	Extend class notation to include the successor function.
class suc A
Definition	df-iord 4103*	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". (Contributed by Jim Kingdon, 10-Oct-2018.) Use its alias dford3 4104 instead for naming consistency with set.mm. (New usage is discouraged.)
|- ( Ord A <-> ( Tr A /\ A. x e. A Tr x ) )
Theorem	dford3 4104*	Alias for df-iord 4103. Use it instead of df-iord 4103 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 4105	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 4106	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 4107 instead for naming consistency with set.mm. (New usage is discouraged.)
|- ( Lim A <-> ( Ord A /\ (/) e. A /\ A = U. A ) )
Theorem	dflim2 4107	Alias for df-ilim 4106. Use it instead of df-ilim 4106 for naming consistency with set.mm. (Contributed by NM, 4-Nov-2004.)
|- ( Lim A <-> ( Ord A /\ (/) e. A /\ A = U. A ) )
Definition	df-suc 4108	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 4149). 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 4109	Equality theorem for the ordinal predicate. (Contributed by NM, 17-Sep-1993.)
|- ( A = B -> ( Ord A <-> Ord B ) )
Theorem	elong 4110	An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)
|- ( A e. V -> ( A e. On <-> Ord A ) )
Theorem	elon 4111	An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)
|- A e. _V   =>    |- ( A e. On <-> Ord A )
Theorem	eloni 4112	An ordinal number has the ordinal property. (Contributed by NM, 5-Jun-1994.)
|- ( A e. On -> Ord A )
Theorem	elon2 4113	An ordinal number is an ordinal set. (Contributed by NM, 8-Feb-2004.)
|- ( A e. On <-> ( Ord A /\ A e. _V ) )
Theorem	limeq 4114	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 4115	An ordinal class is transitive. (Contributed by NM, 3-Apr-1994.)
|- ( Ord A -> Tr A )
Theorem	ordelss 4116	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 4117	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 4118	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 4119	The class of all ordinal numbers is transitive. (Contributed by NM, 4-May-2009.)
|- Tr On
Theorem	ordelon 4120	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 4121	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 4122	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 4123	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 4124	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 4125	Transitive law for ordinal classes. (Contributed by NM, 12-Dec-2004.)
|- ( Ord C -> ( ( A e. B /\ B e. C ) -> A e. C ) )
Theorem	ontr1 4126	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 4127*	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 4128	The empty set is an ordinal class. (Contributed by NM, 11-May-1994.)
|- Ord (/)
Theorem	0elon 4129	The empty set is an ordinal number. Corollary 7N(b) of [Enderton] p. 193. (Contributed by NM, 17-Sep-1993.)
|- (/) e. On
Theorem	inton 4130	The intersection of the class of ordinal numbers is the empty set. (Contributed by NM, 20-Oct-2003.)
|- |^| On = (/)
Theorem	nlim0 4131	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 4132	A limit ordinal is ordinal. (Contributed by NM, 4-May-1995.)
|- ( Lim A -> Ord A )
Theorem	limuni 4133	A limit ordinal is its own supremum (union). (Contributed by NM, 4-May-1995.)
|- ( Lim A -> A = U. A )
Theorem	limuni2 4134	The union of a limit ordinal is a limit ordinal. (Contributed by NM, 19-Sep-2006.)
|- ( Lim A -> Lim U. A )
Theorem	0ellim 4135	A limit ordinal contains the empty set. (Contributed by NM, 15-May-1994.)
|- ( Lim A -> (/) e. A )
Theorem	limelon 4136	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 4137	The class of all ordinal numbers is not empty. (Contributed by NM, 17-Sep-1995.)
|- On =/= (/)
Theorem	onm 4138	The class of all ordinal numbers is inhabited. (Contributed by Jim Kingdon, 6-Mar-2019.)
|- E. x x e. On
Theorem	suceq 4139	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 4140	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 4141	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 4142	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 4143	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 4144	Membership in a successor. (Contributed by NM, 15-Sep-2003.)
|- A e. _V   =>    |- ( B e. suc A <-> ( B e. A \/ B = A ) )
Theorem	nfsuc 4145	Bound-variable hypothesis builder for successor. (Contributed by NM, 15-Sep-2003.)
|- F/_ x A   =>    |- F/_ x suc A
Theorem	elelsuc 4146	Membership in a successor. (Contributed by NM, 20-Jun-1998.)
|- ( A e. B -> A e. suc B )
Theorem	sucel 4147*	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 4148	The successor of the empty set. (Contributed by NM, 1-Feb-2005.)
|- suc (/) = { (/) }
Theorem	sucprc 4149	A proper class is its own successor. (Contributed by NM, 3-Apr-1995.)
|- ( -. A e. _V -> suc A = A )
Theorem	unisuc 4150	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 4151	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 4152	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 4153	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 4154	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 4155	No successor is empty. (Contributed by Jim Kingdon, 14-Oct-2018.)
|- ( A e. V -> suc A =/= (/) )
Theorem	eqelsuc 4156	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 4157*	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 4158	The successor of a transitive class is transitive. (Contributed by Alan Sare, 11-Apr-2009.)
|- ( Tr A -> Tr suc A )
Theorem	trsuc 4159	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 4160	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 4161	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 4162	An ordinal class includes its union. (Contributed by NM, 13-Sep-2003.)
|- ( Ord A -> U. A C_ A )
Theorem	onordi 4163	An ordinal number is an ordinal class. (Contributed by NM, 11-Jun-1994.)
|- A e. On   =>    |- Ord A
Theorem	ontrci 4164	An ordinal number is a transitive class. (Contributed by NM, 11-Jun-1994.)
|- A e. On   =>    |- Tr A
Theorem	oneli 4165	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 4166	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 4167	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 4168	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 4169	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 4170*	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 4172 states that the union itself exists. A version with the standard abbreviation for union is uniex2 4173. A version using class notation is uniex 4174. 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 3878), and (c) the order of the conjuncts is swapped (which is equivalent by ancom 253). The union of a class df-uni 3581 should not be confused with the union of two classes df-un 2922. Their relationship is shown in unipr 3594. (Contributed by NM, 23-Dec-1993.)
|- E. y A. z ( E. w ( z e. w /\ w e. x ) -> z e. y )
Theorem	zfun 4171*	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 4172*	A variant of the Axiom of Union ax-un 4170. 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 4173*	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 4174	The Axiom of Union in class notation. This says that if A is a set i.e. A e. _V (see isset 2561), 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	uniexg 4175	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	unex 4176	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 4177	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 4178	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 4179	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 4180*	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	snnex 4181*	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	opeluu 4182	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 4183*	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 4184*	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 4185*	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 4186*	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 4187*	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 4188*	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 4189*	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 4190*	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 4191*	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 4192*	Two ways to express single-valuedness of a class expression C ( y ). See reusv1 4190 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 4193*	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 4194*	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 4195*	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 4196*	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 4197*	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 ) )
Theorem	ralxfr 4198*	Transfer universal quantification from a variable x to another variable y contained in expression A. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.)
|- ( y e. C -> A e. B )   &    |- ( x e. B -> E. y e. C x = A )   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( A. x e. B ph <-> A. y e. C ps )
Theorem	ralxfrALT 4199*	Transfer universal quantification from a variable x to another variable y contained in expression A. This proof does not use ralxfrd 4194. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( y e. C -> A e. B )   &    |- ( x e. B -> E. y e. C x = A )   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( A. x e. B ph <-> A. y e. C ps )
Theorem	rexxfr 4200*	Transfer existence from a variable x to another variable y contained in expression A. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.)
|- ( y e. C -> A e. B )   &    |- ( x e. B -> E. y e. C x = A )   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( E. x e. B ph <-> E. y e. C ps )