P. 27 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 2601-2700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	r19.26m 2601	Theorem 19.26 of [Margaris] p. 90 with mixed quantifiers. (Contributed by NM, 22-Feb-2004.)
|- ( A. x ( ( x e. A -> ph ) /\ ( x e. B -> ps ) ) <-> ( A. x e. A ph /\ A. x e. B ps ) )
Theorem	ralbi 2602	Distribute a restricted universal quantifier over a biconditional. Theorem 19.15 of [Margaris] p. 90 with restricted quantification. (Contributed by NM, 6-Oct-2003.)
|- ( A. x e. A ( ph <-> ps ) -> ( A. x e. A ph <-> A. x e. A ps ) )
Theorem	rexbi 2603	Distribute a restricted existential quantifier over a biconditional. Theorem 19.18 of [Margaris] p. 90 with restricted quantification. (Contributed by Jim Kingdon, 21-Jan-2019.)
|- ( A. x e. A ( ph <-> ps ) -> ( E. x e. A ph <-> E. x e. A ps ) )
Theorem	ralbiim 2604	Split a biconditional and distribute quantifier. (Contributed by NM, 3-Jun-2012.)
|- ( A. x e. A ( ph <-> ps ) <-> ( A. x e. A ( ph -> ps ) /\ A. x e. A ( ps -> ph ) ) )
Theorem	r19.27av 2605*	Restricted version of one direction of Theorem 19.27 of [Margaris] p. 90. (The other direction doesn't hold when A is empty.) (Contributed by NM, 3-Jun-2004.) (Proof shortened by Andrew Salmon, 30-May-2011.)
|- ( ( A. x e. A ph /\ ps ) -> A. x e. A ( ph /\ ps ) )
Theorem	r19.28av 2606*	Restricted version of one direction of Theorem 19.28 of [Margaris] p. 90. (The other direction doesn't hold when A is empty.) (Contributed by NM, 2-Apr-2004.)
|- ( ( ph /\ A. x e. A ps ) -> A. x e. A ( ph /\ ps ) )
Theorem	r19.29 2607	Theorem 19.29 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 31-Aug-1999.) (Proof shortened by Andrew Salmon, 30-May-2011.)
|- ( ( A. x e. A ph /\ E. x e. A ps ) -> E. x e. A ( ph /\ ps ) )
Theorem	r19.29r 2608	Variation of Theorem 19.29 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 31-Aug-1999.)
|- ( ( E. x e. A ph /\ A. x e. A ps ) -> E. x e. A ( ph /\ ps ) )
Theorem	ralnex2 2609	Relationship between two restricted universal and existential quantifiers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Proof shortened by Wolf Lammen, 18-May-2023.)
|- ( A. x e. A A. y e. B -. ph <-> -. E. x e. A E. y e. B ph )
Theorem	r19.29af2 2610	A commonly used pattern based on r19.29 2607. (Contributed by Thierry Arnoux, 17-Dec-2017.)
|- F/ x ph   &    |- F/ x ch   &    |- ( ( ( ph /\ x e. A ) /\ ps ) -> ch )   &    |- ( ph -> E. x e. A ps )   =>    |- ( ph -> ch )
Theorem	r19.29af 2611*	A commonly used pattern based on r19.29 2607. (Contributed by Thierry Arnoux, 29-Nov-2017.)
|- F/ x ph   &    |- ( ( ( ph /\ x e. A ) /\ ps ) -> ch )   &    |- ( ph -> E. x e. A ps )   =>    |- ( ph -> ch )
Theorem	r19.29an 2612*	A commonly used pattern based on r19.29 2607. (Contributed by Thierry Arnoux, 29-Dec-2019.)
|- ( ( ( ph /\ x e. A ) /\ ps ) -> ch )   =>    |- ( ( ph /\ E. x e. A ps ) -> ch )
Theorem	r19.29a 2613*	A commonly used pattern based on r19.29 2607. (Contributed by Thierry Arnoux, 22-Nov-2017.)
|- ( ( ( ph /\ x e. A ) /\ ps ) -> ch )   &    |- ( ph -> E. x e. A ps )   =>    |- ( ph -> ch )
Theorem	r19.29d2r 2614	Theorem 19.29 of [Margaris] p. 90 with two restricted quantifiers, deduction version. (Contributed by Thierry Arnoux, 30-Jan-2017.)
|- ( ph -> A. x e. A A. y e. B ps )   &    |- ( ph -> E. x e. A E. y e. B ch )   =>    |- ( ph -> E. x e. A E. y e. B ( ps /\ ch ) )
Theorem	r19.29vva 2615*	A commonly used pattern based on r19.29 2607, version with two restricted quantifiers. (Contributed by Thierry Arnoux, 26-Nov-2017.)
|- ( ( ( ( ph /\ x e. A ) /\ y e. B ) /\ ps ) -> ch )   &    |- ( ph -> E. x e. A E. y e. B ps )   =>    |- ( ph -> ch )
Theorem	r19.32r 2616	One direction of Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers. For decidable propositions this is an equivalence. (Contributed by Jim Kingdon, 19-Aug-2018.)
|- F/ x ph   =>    |- ( ( ph \/ A. x e. A ps ) -> A. x e. A ( ph \/ ps ) )
Theorem	r19.30dc 2617	Restricted quantifier version of 19.30dc 1620. (Contributed by Scott Fenton, 25-Feb-2011.) (Proof shortened by Wolf Lammen, 18-Jun-2023.)
|- ( ( A. x e. A ( ph \/ ps ) /\ DECID E. x e. A ps ) -> ( A. x e. A ph \/ E. x e. A ps ) )
Theorem	r19.32vr 2618*	One direction of Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers. For decidable propositions this is an equivalence, as seen at r19.32vdc 2619. (Contributed by Jim Kingdon, 19-Aug-2018.)
|- ( ( ph \/ A. x e. A ps ) -> A. x e. A ( ph \/ ps ) )
Theorem	r19.32vdc 2619*	Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers, where ph is decidable. (Contributed by Jim Kingdon, 4-Jun-2018.)
|- (DECID ph -> ( A. x e. A ( ph \/ ps ) <-> ( ph \/ A. x e. A ps ) ) )
Theorem	r19.35-1 2620	Restricted quantifier version of 19.35-1 1617. (Contributed by Jim Kingdon, 4-Jun-2018.)
|- ( E. x e. A ( ph -> ps ) -> ( A. x e. A ph -> E. x e. A ps ) )
Theorem	r19.36av 2621*	One direction of a restricted quantifier version of Theorem 19.36 of [Margaris] p. 90. In classical logic, the converse would hold if A has at least one element, but in intuitionistic logic, that is not a sufficient condition. (Contributed by NM, 22-Oct-2003.)
|- ( E. x e. A ( ph -> ps ) -> ( A. x e. A ph -> ps ) )
Theorem	r19.37 2622	Restricted version of one direction of Theorem 19.37 of [Margaris] p. 90. In classical logic the converse would hold if A has at least one element, but that is not sufficient in intuitionistic logic. (Contributed by FL, 13-May-2012.) (Revised by Mario Carneiro, 11-Dec-2016.)
|- F/ x ph   =>    |- ( E. x e. A ( ph -> ps ) -> ( ph -> E. x e. A ps ) )
Theorem	r19.37av 2623*	Restricted version of one direction of Theorem 19.37 of [Margaris] p. 90. (Contributed by NM, 2-Apr-2004.)
|- ( E. x e. A ( ph -> ps ) -> ( ph -> E. x e. A ps ) )
Theorem	r19.40 2624	Restricted quantifier version of Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 2-Apr-2004.)
|- ( E. x e. A ( ph /\ ps ) -> ( E. x e. A ph /\ E. x e. A ps ) )
Theorem	r19.41 2625	Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 1-Nov-2010.)
|- F/ x ps   =>    |- ( E. x e. A ( ph /\ ps ) <-> ( E. x e. A ph /\ ps ) )
Theorem	r19.41v 2626*	Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 17-Dec-2003.)
|- ( E. x e. A ( ph /\ ps ) <-> ( E. x e. A ph /\ ps ) )
Theorem	r19.42v 2627*	Restricted version of Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
|- ( E. x e. A ( ph /\ ps ) <-> ( ph /\ E. x e. A ps ) )
Theorem	r19.43 2628	Restricted version of Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.) (Proof rewritten by Jim Kingdon, 5-Jun-2018.)
|- ( E. x e. A ( ph \/ ps ) <-> ( E. x e. A ph \/ E. x e. A ps ) )
Theorem	r19.44av 2629*	One direction of a restricted quantifier version of Theorem 19.44 of [Margaris] p. 90. The other direction doesn't hold when A is empty. (Contributed by NM, 2-Apr-2004.)
|- ( E. x e. A ( ph \/ ps ) -> ( E. x e. A ph \/ ps ) )
Theorem	r19.45av 2630*	Restricted version of one direction of Theorem 19.45 of [Margaris] p. 90. (The other direction doesn't hold when A is empty.) (Contributed by NM, 2-Apr-2004.)
|- ( E. x e. A ( ph \/ ps ) -> ( ph \/ E. x e. A ps ) )
Theorem	ralcomf 2631*	Commutation of restricted quantifiers. (Contributed by Mario Carneiro, 14-Oct-2016.)
|- F/_ y A   &    |- F/_ x B   =>    |- ( A. x e. A A. y e. B ph <-> A. y e. B A. x e. A ph )
Theorem	rexcomf 2632*	Commutation of restricted quantifiers. (Contributed by Mario Carneiro, 14-Oct-2016.)
|- F/_ y A   &    |- F/_ x B   =>    |- ( E. x e. A E. y e. B ph <-> E. y e. B E. x e. A ph )
Theorem	ralcom 2633*	Commutation of restricted quantifiers. (Contributed by NM, 13-Oct-1999.) (Revised by Mario Carneiro, 14-Oct-2016.)
|- ( A. x e. A A. y e. B ph <-> A. y e. B A. x e. A ph )
Theorem	rexcom 2634*	Commutation of restricted quantifiers. (Contributed by NM, 19-Nov-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
|- ( E. x e. A E. y e. B ph <-> E. y e. B E. x e. A ph )
Theorem	rexcom13 2635*	Swap 1st and 3rd restricted existential quantifiers. (Contributed by NM, 8-Apr-2015.)
|- ( E. x e. A E. y e. B E. z e. C ph <-> E. z e. C E. y e. B E. x e. A ph )
Theorem	rexrot4 2636*	Rotate existential restricted quantifiers twice. (Contributed by NM, 8-Apr-2015.)
|- ( E. x e. A E. y e. B E. z e. C E. w e. D ph <-> E. z e. C E. w e. D E. x e. A E. y e. B ph )
Theorem	ralcom3 2637	A commutative law for restricted quantifiers that swaps the domain of the restriction. (Contributed by NM, 22-Feb-2004.)
|- ( A. x e. A ( x e. B -> ph ) <-> A. x e. B ( x e. A -> ph ) )
Theorem	reean 2638*	Rearrange existential quantifiers. (Contributed by NM, 27-Oct-2010.) (Proof shortened by Andrew Salmon, 30-May-2011.)
|- F/ y ph   &    |- F/ x ps   =>    |- ( E. x e. A E. y e. B ( ph /\ ps ) <-> ( E. x e. A ph /\ E. y e. B ps ) )
Theorem	reeanv 2639*	Rearrange existential quantifiers. (Contributed by NM, 9-May-1999.)
|- ( E. x e. A E. y e. B ( ph /\ ps ) <-> ( E. x e. A ph /\ E. y e. B ps ) )
Theorem	3reeanv 2640*	Rearrange three existential quantifiers. (Contributed by Jeff Madsen, 11-Jun-2010.)
|- ( E. x e. A E. y e. B E. z e. C ( ph /\ ps /\ ch ) <-> ( E. x e. A ph /\ E. y e. B ps /\ E. z e. C ch ) )
Theorem	nfreu1 2641	x is not free in E! x e. A ph. (Contributed by NM, 19-Mar-1997.)
|- F/ x E! x e. A ph
Theorem	nfrmo1 2642	x is not free in E* x e. A ph. (Contributed by NM, 16-Jun-2017.)
|- F/ x E* x e. A ph
Theorem	nfreudxy 2643*	Not-free deduction for restricted uniqueness. This is a version where x and y are distinct. (Contributed by Jim Kingdon, 6-Jun-2018.)
|- F/ y ph   &    |- ( ph -> F/_ x A )   &    |- ( ph -> F/ x ps )   =>    |- ( ph -> F/ x E! y e. A ps )
Theorem	nfreuxy 2644*	Not-free for restricted uniqueness. This is a version where x and y are distinct. (Contributed by Jim Kingdon, 6-Jun-2018.)
|- F/_ x A   &    |- F/ x ph   =>    |- F/ x E! y e. A ph
Theorem	rabid 2645	An "identity" law of concretion for restricted abstraction. Special case of Definition 2.1 of [Quine] p. 16. (Contributed by NM, 9-Oct-2003.)
|- ( x e. { x e. A | ph } <-> ( x e. A /\ ph ) )
Theorem	rabid2 2646*	An "identity" law for restricted class abstraction. (Contributed by NM, 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)
|- ( A = { x e. A | ph } <-> A. x e. A ph )
Theorem	rabbi 2647	Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbidva 2718. (Contributed by NM, 25-Nov-2013.)
|- ( A. x e. A ( ps <-> ch ) <-> { x e. A | ps } = { x e. A | ch } )
Theorem	rabswap 2648	Swap with a membership relation in a restricted class abstraction. (Contributed by NM, 4-Jul-2005.)
|- { x e. A | x e. B } = { x e. B | x e. A }
Theorem	nfrab1 2649	The abstraction variable in a restricted class abstraction isn't free. (Contributed by NM, 19-Mar-1997.)
|- F/_ x { x e. A | ph }
Theorem	nfrabxy 2650*	A variable not free in a wff remains so in a restricted class abstraction. (Contributed by Jim Kingdon, 19-Jul-2018.)
|- F/ x ph   &    |- F/_ x A   =>    |- F/_ x { y e. A | ph }
Theorem	reubida 2651	Formula-building rule for restricted existential quantifier (deduction form). (Contributed by Mario Carneiro, 19-Nov-2016.)
|- F/ x ph   &    |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( E! x e. A ps <-> E! x e. A ch ) )
Theorem	reubidva 2652*	Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 13-Nov-2004.)
|- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( E! x e. A ps <-> E! x e. A ch ) )
Theorem	reubidv 2653*	Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 17-Oct-1996.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( E! x e. A ps <-> E! x e. A ch ) )
Theorem	reubiia 2654	Formula-building rule for restricted existential quantifier (inference form). (Contributed by NM, 14-Nov-2004.)
|- ( x e. A -> ( ph <-> ps ) )   =>    |- ( E! x e. A ph <-> E! x e. A ps )
Theorem	reubii 2655	Formula-building rule for restricted existential quantifier (inference form). (Contributed by NM, 22-Oct-1999.)
|- ( ph <-> ps )   =>    |- ( E! x e. A ph <-> E! x e. A ps )
Theorem	rmobida 2656	Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 16-Jun-2017.)
|- F/ x ph   &    |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( E* x e. A ps <-> E* x e. A ch ) )
Theorem	rmobidva 2657*	Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 16-Jun-2017.)
|- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( E* x e. A ps <-> E* x e. A ch ) )
Theorem	rmobidv 2658*	Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 16-Jun-2017.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( E* x e. A ps <-> E* x e. A ch ) )
Theorem	rmobiia 2659	Formula-building rule for restricted existential quantifier (inference form). (Contributed by NM, 16-Jun-2017.)
|- ( x e. A -> ( ph <-> ps ) )   =>    |- ( E* x e. A ph <-> E* x e. A ps )
Theorem	rmobii 2660	Formula-building rule for restricted existential quantifier (inference form). (Contributed by NM, 16-Jun-2017.)
|- ( ph <-> ps )   =>    |- ( E* x e. A ph <-> E* x e. A ps )
Theorem	raleqf 2661	Equality theorem for restricted universal quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.) (Revised by Andrew Salmon, 11-Jul-2011.)
|- F/_ x A   &    |- F/_ x B   =>    |- ( A = B -> ( A. x e. A ph <-> A. x e. B ph ) )
Theorem	rexeqf 2662	Equality theorem for restricted existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 9-Oct-2003.) (Revised by Andrew Salmon, 11-Jul-2011.)
|- F/_ x A   &    |- F/_ x B   =>    |- ( A = B -> ( E. x e. A ph <-> E. x e. B ph ) )
Theorem	reueq1f 2663	Equality theorem for restricted unique existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 5-Apr-2004.) (Revised by Andrew Salmon, 11-Jul-2011.)
|- F/_ x A   &    |- F/_ x B   =>    |- ( A = B -> ( E! x e. A ph <-> E! x e. B ph ) )
Theorem	rmoeq1f 2664	Equality theorem for restricted at-most-one quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
|- F/_ x A   &    |- F/_ x B   =>    |- ( A = B -> ( E* x e. A ph <-> E* x e. B ph ) )
Theorem	raleq 2665*	Equality theorem for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.)
|- ( A = B -> ( A. x e. A ph <-> A. x e. B ph ) )
Theorem	rexeq 2666*	Equality theorem for restricted existential quantifier. (Contributed by NM, 29-Oct-1995.)
|- ( A = B -> ( E. x e. A ph <-> E. x e. B ph ) )
Theorem	reueq1 2667*	Equality theorem for restricted unique existential quantifier. (Contributed by NM, 5-Apr-2004.)
|- ( A = B -> ( E! x e. A ph <-> E! x e. B ph ) )
Theorem	rmoeq1 2668*	Equality theorem for restricted at-most-one quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
|- ( A = B -> ( E* x e. A ph <-> E* x e. B ph ) )
Theorem	raleqi 2669*	Equality inference for restricted universal qualifier. (Contributed by Paul Chapman, 22-Jun-2011.)
|- A = B   =>    |- ( A. x e. A ph <-> A. x e. B ph )
Theorem	rexeqi 2670*	Equality inference for restricted existential qualifier. (Contributed by Mario Carneiro, 23-Apr-2015.)
|- A = B   =>    |- ( E. x e. A ph <-> E. x e. B ph )
Theorem	raleqdv 2671*	Equality deduction for restricted universal quantifier. (Contributed by NM, 13-Nov-2005.)
|- ( ph -> A = B )   =>    |- ( ph -> ( A. x e. A ps <-> A. x e. B ps ) )
Theorem	rexeqdv 2672*	Equality deduction for restricted existential quantifier. (Contributed by NM, 14-Jan-2007.)
|- ( ph -> A = B )   =>    |- ( ph -> ( E. x e. A ps <-> E. x e. B ps ) )
Theorem	raleqbi1dv 2673*	Equality deduction for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.)
|- ( A = B -> ( ph <-> ps ) )   =>    |- ( A = B -> ( A. x e. A ph <-> A. x e. B ps ) )
Theorem	rexeqbi1dv 2674*	Equality deduction for restricted existential quantifier. (Contributed by NM, 18-Mar-1997.)
|- ( A = B -> ( ph <-> ps ) )   =>    |- ( A = B -> ( E. x e. A ph <-> E. x e. B ps ) )
Theorem	reueqd 2675*	Equality deduction for restricted unique existential quantifier. (Contributed by NM, 5-Apr-2004.)
|- ( A = B -> ( ph <-> ps ) )   =>    |- ( A = B -> ( E! x e. A ph <-> E! x e. B ps ) )
Theorem	rmoeqd 2676*	Equality deduction for restricted at-most-one quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
|- ( A = B -> ( ph <-> ps ) )   =>    |- ( A = B -> ( E* x e. A ph <-> E* x e. B ps ) )
Theorem	raleqbidv 2677*	Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.)
|- ( ph -> A = B )   &    |- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( A. x e. A ps <-> A. x e. B ch ) )
Theorem	rexeqbidv 2678*	Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.)
|- ( ph -> A = B )   &    |- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( E. x e. A ps <-> E. x e. B ch ) )
Theorem	raleqbidva 2679*	Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)
|- ( ph -> A = B )   &    |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( A. x e. A ps <-> A. x e. B ch ) )
Theorem	rexeqbidva 2680*	Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)
|- ( ph -> A = B )   &    |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( E. x e. A ps <-> E. x e. B ch ) )
Theorem	mormo 2681	Unrestricted "at most one" implies restricted "at most one". (Contributed by NM, 16-Jun-2017.)
|- ( E* x ph -> E* x e. A ph )
Theorem	reu5 2682	Restricted uniqueness in terms of "at most one". (Contributed by NM, 23-May-1999.) (Revised by NM, 16-Jun-2017.)
|- ( E! x e. A ph <-> ( E. x e. A ph /\ E* x e. A ph ) )
Theorem	reurex 2683	Restricted unique existence implies restricted existence. (Contributed by NM, 19-Aug-1999.)
|- ( E! x e. A ph -> E. x e. A ph )
Theorem	reurmo 2684	Restricted existential uniqueness implies restricted "at most one." (Contributed by NM, 16-Jun-2017.)
|- ( E! x e. A ph -> E* x e. A ph )
Theorem	rmo5 2685	Restricted "at most one" in term of uniqueness. (Contributed by NM, 16-Jun-2017.)
|- ( E* x e. A ph <-> ( E. x e. A ph -> E! x e. A ph ) )
Theorem	nrexrmo 2686	Nonexistence implies restricted "at most one". (Contributed by NM, 17-Jun-2017.)
|- ( -. E. x e. A ph -> E* x e. A ph )
Theorem	cbvralfw 2687*	Rule used to change bound variables, using implicit substitution. Version of cbvralf 2689 with a disjoint variable condition. Although we don't do so yet, we expect this disjoint variable condition will allow us to remove reliance on ax-i12 1500 and ax-bndl 1502 in the proof. (Contributed by NM, 7-Mar-2004.) (Revised by Gino Giotto, 23-May-2024.)
|- F/_ x A   &    |- F/_ y A   &    |- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( A. x e. A ph <-> A. y e. A ps )
Theorem	cbvrexfw 2688*	Rule used to change bound variables, using implicit substitution. Version of cbvrexf 2690 with a disjoint variable condition, which does not require ax-13 2143. (Contributed by FL, 27-Apr-2008.) (Revised by Gino Giotto, 10-Jan-2024.)
|- F/_ x A   &    |- F/_ y A   &    |- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( E. x e. A ph <-> E. y e. A ps )
Theorem	cbvralf 2689	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 7-Mar-2004.) (Revised by Mario Carneiro, 9-Oct-2016.)
|- F/_ x A   &    |- F/_ y A   &    |- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( A. x e. A ph <-> A. y e. A ps )
Theorem	cbvrexf 2690	Rule used to change bound variables, using implicit substitution. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 9-Oct-2016.) (Proof rewritten by Jim Kingdon, 10-Jun-2018.)
|- F/_ x A   &    |- F/_ y A   &    |- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( E. x e. A ph <-> E. y e. A ps )
Theorem	cbvralw 2691*	Rule used to change bound variables, using implicit substitution. Version of cbvral 2692 with a disjoint variable condition. Although we don't do so yet, we expect this disjoint variable condition will allow us to remove reliance on ax-i12 1500 and ax-bndl 1502 in the proof. (Contributed by NM, 31-Jul-2003.) (Revised by Gino Giotto, 10-Jan-2024.)
|- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( A. x e. A ph <-> A. y e. A ps )
Theorem	cbvral 2692*	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.)
|- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( A. x e. A ph <-> A. y e. A ps )
Theorem	cbvrex 2693*	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
|- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( E. x e. A ph <-> E. y e. A ps )
Theorem	cbvreu 2694*	Change the bound variable of a restricted unique existential quantifier using implicit substitution. (Contributed by Mario Carneiro, 15-Oct-2016.)
|- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( E! x e. A ph <-> E! y e. A ps )
Theorem	cbvrmo 2695*	Change the bound variable of restricted "at most one" using implicit substitution. (Contributed by NM, 16-Jun-2017.)
|- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( E* x e. A ph <-> E* y e. A ps )
Theorem	cbvralv 2696*	Change the bound variable of a restricted universal quantifier using implicit substitution. (Contributed by NM, 28-Jan-1997.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( A. x e. A ph <-> A. y e. A ps )
Theorem	cbvrexv 2697*	Change the bound variable of a restricted existential quantifier using implicit substitution. (Contributed by NM, 2-Jun-1998.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( E. x e. A ph <-> E. y e. A ps )
Theorem	cbvreuv 2698*	Change the bound variable of a restricted unique existential quantifier using implicit substitution. (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( E! x e. A ph <-> E! y e. A ps )
Theorem	cbvrmov 2699*	Change the bound variable of a restricted at-most-one quantifier using implicit substitution. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( E* x e. A ph <-> E* y e. A ps )
Theorem	cbvralvw 2700*	Version of cbvralv 2696 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) Reduce axiom usage. (Revised by Gino Giotto, 25-Aug-2024.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( A. x e. A ph <-> A. y e. A ps )