P. 28 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 2701-2800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ceqsex6v 2701*	Elimination of six existential quantifiers, using implicit substitution. (Contributed by NM, 21-Sep-2011.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- D e. _V   &    |- E e. _V   &    |- F e. _V   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ( y = B -> ( ps <-> ch ) )   &    |- ( z = C -> ( ch <-> th ) )   &    |- ( w = D -> ( th <-> ta ) )   &    |- ( v = E -> ( ta <-> et ) )   &    |- ( u = F -> ( et <-> ze ) )   =>    |- ( E. x E. y E. z E. w E. v E. u ( ( x = A /\ y = B /\ z = C ) /\ ( w = D /\ v = E /\ u = F ) /\ ph ) <-> ze )
Theorem	ceqsex8v 2702*	Elimination of eight existential quantifiers, using implicit substitution. (Contributed by NM, 23-Sep-2011.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- D e. _V   &    |- E e. _V   &    |- F e. _V   &    |- G e. _V   &    |- H e. _V   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ( y = B -> ( ps <-> ch ) )   &    |- ( z = C -> ( ch <-> th ) )   &    |- ( w = D -> ( th <-> ta ) )   &    |- ( v = E -> ( ta <-> et ) )   &    |- ( u = F -> ( et <-> ze ) )   &    |- ( t = G -> ( ze <-> si ) )   &    |- ( s = H -> ( si <-> rh ) )   =>    |- ( E. x E. y E. z E. w E. v E. u E. t E. s ( ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) /\ ( ( v = E /\ u = F ) /\ ( t = G /\ s = H ) ) /\ ph ) <-> rh )
Theorem	gencbvex 2703*	Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
|- A e. _V   &    |- ( A = y -> ( ph <-> ps ) )   &    |- ( A = y -> ( ch <-> th ) )   &    |- ( th <-> E. x ( ch /\ A = y ) )   =>    |- ( E. x ( ch /\ ph ) <-> E. y ( th /\ ps ) )
Theorem	gencbvex2 2704*	Restatement of gencbvex 2703 with weaker hypotheses. (Contributed by Jeff Hankins, 6-Dec-2006.)
|- A e. _V   &    |- ( A = y -> ( ph <-> ps ) )   &    |- ( A = y -> ( ch <-> th ) )   &    |- ( th -> E. x ( ch /\ A = y ) )   =>    |- ( E. x ( ch /\ ph ) <-> E. y ( th /\ ps ) )
Theorem	gencbval 2705*	Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.) (Proof rewritten by Jim Kingdon, 20-Jun-2018.)
|- A e. _V   &    |- ( A = y -> ( ph <-> ps ) )   &    |- ( A = y -> ( ch <-> th ) )   &    |- ( th <-> E. x ( ch /\ A = y ) )   =>    |- ( A. x ( ch -> ph ) <-> A. y ( th -> ps ) )
Theorem	sbhypf 2706*	Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf . (Contributed by Raph Levien, 10-Apr-2004.)
|- F/ x ps   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( y = A -> ( [ y / x ] ph <-> ps ) )
Theorem	vtoclgft 2707	Closed theorem form of vtoclgf 2715. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 12-Oct-2016.)
|- ( ( ( F/_ x A /\ F/ x ps ) /\ ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) /\ A e. V ) -> ps )
Theorem	vtocldf 2708	Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.)
|- ( ph -> A e. V )   &    |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )   &    |- ( ph -> ps )   &    |- F/ x ph   &    |- ( ph -> F/_ x A )   &    |- ( ph -> F/ x ch )   =>    |- ( ph -> ch )
Theorem	vtocld 2709*	Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.)
|- ( ph -> A e. V )   &    |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )   &    |- ( ph -> ps )   =>    |- ( ph -> ch )
Theorem	vtoclf 2710*	Implicit substitution of a class for a setvar variable. This is a generalization of chvar 1713. (Contributed by NM, 30-Aug-1993.)
|- F/ x ps   &    |- A e. _V   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ph   =>    |- ps
Theorem	vtocl 2711*	Implicit substitution of a class for a setvar variable. (Contributed by NM, 30-Aug-1993.)
|- A e. _V   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ph   =>    |- ps
Theorem	vtocl2 2712*	Implicit substitution of classes for setvar variables. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
|- A e. _V   &    |- B e. _V   &    |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )   &    |- ph   =>    |- ps
Theorem	vtocl3 2713*	Implicit substitution of classes for setvar variables. (Contributed by NM, 3-Jun-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> ps ) )   &    |- ph   =>    |- ps
Theorem	vtoclb 2714*	Implicit substitution of a class for a setvar variable. (Contributed by NM, 23-Dec-1993.)
|- A e. _V   &    |- ( x = A -> ( ph <-> ch ) )   &    |- ( x = A -> ( ps <-> th ) )   &    |- ( ph <-> ps )   =>    |- ( ch <-> th )
Theorem	vtoclgf 2715	Implicit substitution of a class for a setvar variable, with bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.) (Proof shortened by Mario Carneiro, 10-Oct-2016.)
|- F/_ x A   &    |- F/ x ps   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ph   =>    |- ( A e. V -> ps )
Theorem	vtoclg1f 2716*	Version of vtoclgf 2715 with one non-freeness hypothesis replaced with a disjoint variable condition, thus avoiding dependency on ax-11 1467 and ax-13 1474. (Contributed by BJ, 1-May-2019.)
|- F/ x ps   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ph   =>    |- ( A e. V -> ps )
Theorem	vtoclg 2717*	Implicit substitution of a class expression for a setvar variable. (Contributed by NM, 17-Apr-1995.)
|- ( x = A -> ( ph <-> ps ) )   &    |- ph   =>    |- ( A e. V -> ps )
Theorem	vtoclbg 2718*	Implicit substitution of a class for a setvar variable. (Contributed by NM, 29-Apr-1994.)
|- ( x = A -> ( ph <-> ch ) )   &    |- ( x = A -> ( ps <-> th ) )   &    |- ( ph <-> ps )   =>    |- ( A e. V -> ( ch <-> th ) )
Theorem	vtocl2gf 2719	Implicit substitution of a class for a setvar variable. (Contributed by NM, 25-Apr-1995.)
|- F/_ x A   &    |- F/_ y A   &    |- F/_ y B   &    |- F/ x ps   &    |- F/ y ch   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ( y = B -> ( ps <-> ch ) )   &    |- ph   =>    |- ( ( A e. V /\ B e. W ) -> ch )
Theorem	vtocl3gf 2720	Implicit substitution of a class for a setvar variable. (Contributed by NM, 10-Aug-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
|- F/_ x A   &    |- F/_ y A   &    |- F/_ z A   &    |- F/_ y B   &    |- F/_ z B   &    |- F/_ z C   &    |- F/ x ps   &    |- F/ y ch   &    |- F/ z th   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ( y = B -> ( ps <-> ch ) )   &    |- ( z = C -> ( ch <-> th ) )   &    |- ph   =>    |- ( ( A e. V /\ B e. W /\ C e. X ) -> th )
Theorem	vtocl2g 2721*	Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 25-Apr-1995.)
|- ( x = A -> ( ph <-> ps ) )   &    |- ( y = B -> ( ps <-> ch ) )   &    |- ph   =>    |- ( ( A e. V /\ B e. W ) -> ch )
Theorem	vtoclgaf 2722*	Implicit substitution of a class for a setvar variable. (Contributed by NM, 17-Feb-2006.) (Revised by Mario Carneiro, 10-Oct-2016.)
|- F/_ x A   &    |- F/ x ps   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ( x e. B -> ph )   =>    |- ( A e. B -> ps )
Theorem	vtoclga 2723*	Implicit substitution of a class for a setvar variable. (Contributed by NM, 20-Aug-1995.)
|- ( x = A -> ( ph <-> ps ) )   &    |- ( x e. B -> ph )   =>    |- ( A e. B -> ps )
Theorem	vtocl2gaf 2724*	Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 10-Aug-2013.)
|- F/_ x A   &    |- F/_ y A   &    |- F/_ y B   &    |- F/ x ps   &    |- F/ y ch   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ( y = B -> ( ps <-> ch ) )   &    |- ( ( x e. C /\ y e. D ) -> ph )   =>    |- ( ( A e. C /\ B e. D ) -> ch )
Theorem	vtocl2ga 2725*	Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 20-Aug-1995.)
|- ( x = A -> ( ph <-> ps ) )   &    |- ( y = B -> ( ps <-> ch ) )   &    |- ( ( x e. C /\ y e. D ) -> ph )   =>    |- ( ( A e. C /\ B e. D ) -> ch )
Theorem	vtocl3gaf 2726*	Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 10-Aug-2013.) (Revised by Mario Carneiro, 11-Oct-2016.)
|- F/_ x A   &    |- F/_ y A   &    |- F/_ z A   &    |- F/_ y B   &    |- F/_ z B   &    |- F/_ z C   &    |- F/ x ps   &    |- F/ y ch   &    |- F/ z th   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ( y = B -> ( ps <-> ch ) )   &    |- ( z = C -> ( ch <-> th ) )   &    |- ( ( x e. R /\ y e. S /\ z e. T ) -> ph )   =>    |- ( ( A e. R /\ B e. S /\ C e. T ) -> th )
Theorem	vtocl3ga 2727*	Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 20-Aug-1995.)
|- ( x = A -> ( ph <-> ps ) )   &    |- ( y = B -> ( ps <-> ch ) )   &    |- ( z = C -> ( ch <-> th ) )   &    |- ( ( x e. D /\ y e. R /\ z e. S ) -> ph )   =>    |- ( ( A e. D /\ B e. R /\ C e. S ) -> th )
Theorem	vtocleg 2728*	Implicit substitution of a class for a setvar variable. (Contributed by NM, 10-Jan-2004.)
|- ( x = A -> ph )   =>    |- ( A e. V -> ph )
Theorem	vtoclegft 2729*	Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef 2730.) (Contributed by NM, 7-Nov-2005.) (Revised by Mario Carneiro, 11-Oct-2016.)
|- ( ( A e. B /\ F/ x ph /\ A. x ( x = A -> ph ) ) -> ph )
Theorem	vtoclef 2730*	Implicit substitution of a class for a setvar variable. (Contributed by NM, 18-Aug-1993.)
|- F/ x ph   &    |- A e. _V   &    |- ( x = A -> ph )   =>    |- ph
Theorem	vtocle 2731*	Implicit substitution of a class for a setvar variable. (Contributed by NM, 9-Sep-1993.)
|- A e. _V   &    |- ( x = A -> ph )   =>    |- ph
Theorem	vtoclri 2732*	Implicit substitution of a class for a setvar variable. (Contributed by NM, 21-Nov-1994.)
|- ( x = A -> ( ph <-> ps ) )   &    |- A. x e. B ph   =>    |- ( A e. B -> ps )
Theorem	spcimgft 2733	A closed version of spcimgf 2737. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- F/ x ps   &    |- F/_ x A   =>    |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( A e. B -> ( A. x ph -> ps ) ) )
Theorem	spcgft 2734	A closed version of spcgf 2739. (Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro, 4-Jan-2017.)
|- F/ x ps   &    |- F/_ x A   =>    |- ( A. x ( x = A -> ( ph <-> ps ) ) -> ( A e. B -> ( A. x ph -> ps ) ) )
Theorem	spcimegft 2735	A closed version of spcimegf 2738. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- F/ x ps   &    |- F/_ x A   =>    |- ( A. x ( x = A -> ( ps -> ph ) ) -> ( A e. B -> ( ps -> E. x ph ) ) )
Theorem	spcegft 2736	A closed version of spcegf 2740. (Contributed by Jim Kingdon, 22-Jun-2018.)
|- F/ x ps   &    |- F/_ x A   =>    |- ( A. x ( x = A -> ( ph <-> ps ) ) -> ( A e. B -> ( ps -> E. x ph ) ) )
Theorem	spcimgf 2737	Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- F/_ x A   &    |- F/ x ps   &    |- ( x = A -> ( ph -> ps ) )   =>    |- ( A e. V -> ( A. x ph -> ps ) )
Theorem	spcimegf 2738	Existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- F/_ x A   &    |- F/ x ps   &    |- ( x = A -> ( ps -> ph ) )   =>    |- ( A e. V -> ( ps -> E. x ph ) )
Theorem	spcgf 2739	Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 2-Feb-1997.) (Revised by Andrew Salmon, 12-Aug-2011.)
|- F/_ x A   &    |- F/ x ps   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. V -> ( A. x ph -> ps ) )
Theorem	spcegf 2740	Existential specialization, using implicit substitution. (Contributed by NM, 2-Feb-1997.)
|- F/_ x A   &    |- F/ x ps   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. V -> ( ps -> E. x ph ) )
Theorem	spcimdv 2741*	Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- ( ph -> A e. B )   &    |- ( ( ph /\ x = A ) -> ( ps -> ch ) )   =>    |- ( ph -> ( A. x ps -> ch ) )
Theorem	spcdv 2742*	Rule of specialization, using implicit substitution. Analogous to rspcdv 2763. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. B )   &    |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( A. x ps -> ch ) )
Theorem	spcimedv 2743*	Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- ( ph -> A e. B )   &    |- ( ( ph /\ x = A ) -> ( ch -> ps ) )   =>    |- ( ph -> ( ch -> E. x ps ) )
Theorem	spcgv 2744*	Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 22-Jun-1994.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. V -> ( A. x ph -> ps ) )
Theorem	spcegv 2745*	Existential specialization, using implicit substitution. (Contributed by NM, 14-Aug-1994.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. V -> ( ps -> E. x ph ) )
Theorem	spc2egv 2746*	Existential specialization with 2 quantifiers, using implicit substitution. (Contributed by NM, 3-Aug-1995.)
|- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )   =>    |- ( ( A e. V /\ B e. W ) -> ( ps -> E. x E. y ph ) )
Theorem	spc2gv 2747*	Specialization with 2 quantifiers, using implicit substitution. (Contributed by NM, 27-Apr-2004.)
|- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )   =>    |- ( ( A e. V /\ B e. W ) -> ( A. x A. y ph -> ps ) )
Theorem	spc3egv 2748*	Existential specialization with 3 quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.)
|- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> ps ) )   =>    |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( ps -> E. x E. y E. z ph ) )
Theorem	spc3gv 2749*	Specialization with 3 quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.)
|- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> ps ) )   =>    |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( A. x A. y A. z ph -> ps ) )
Theorem	spcv 2750*	Rule of specialization, using implicit substitution. (Contributed by NM, 22-Jun-1994.)
|- A e. _V   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( A. x ph -> ps )
Theorem	spcev 2751*	Existential specialization, using implicit substitution. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.)
|- A e. _V   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( ps -> E. x ph )
Theorem	spc2ev 2752*	Existential specialization, using implicit substitution. (Contributed by NM, 3-Aug-1995.)
|- A e. _V   &    |- B e. _V   &    |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )   =>    |- ( ps -> E. x E. y ph )
Theorem	rspct 2753*	A closed version of rspc 2754. (Contributed by Andrew Salmon, 6-Jun-2011.)
|- F/ x ps   =>    |- ( A. x ( x = A -> ( ph <-> ps ) ) -> ( A e. B -> ( A. x e. B ph -> ps ) ) )
Theorem	rspc 2754*	Restricted specialization, using implicit substitution. (Contributed by NM, 19-Apr-2005.) (Revised by Mario Carneiro, 11-Oct-2016.)
|- F/ x ps   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. B -> ( A. x e. B ph -> ps ) )
Theorem	rspce 2755*	Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) (Revised by Mario Carneiro, 11-Oct-2016.)
|- F/ x ps   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( ( A e. B /\ ps ) -> E. x e. B ph )
Theorem	rspcv 2756*	Restricted specialization, using implicit substitution. (Contributed by NM, 26-May-1998.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. B -> ( A. x e. B ph -> ps ) )
Theorem	rspccv 2757*	Restricted specialization, using implicit substitution. (Contributed by NM, 2-Feb-2006.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( A. x e. B ph -> ( A e. B -> ps ) )
Theorem	rspcva 2758*	Restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-2005.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( ( A e. B /\ A. x e. B ph ) -> ps )
Theorem	rspccva 2759*	Restricted specialization, using implicit substitution. (Contributed by NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( ( A. x e. B ph /\ A e. B ) -> ps )
Theorem	rspcev 2760*	Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( ( A e. B /\ ps ) -> E. x e. B ph )
Theorem	rspcimdv 2761*	Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- ( ph -> A e. B )   &    |- ( ( ph /\ x = A ) -> ( ps -> ch ) )   =>    |- ( ph -> ( A. x e. B ps -> ch ) )
Theorem	rspcimedv 2762*	Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- ( ph -> A e. B )   &    |- ( ( ph /\ x = A ) -> ( ch -> ps ) )   =>    |- ( ph -> ( ch -> E. x e. B ps ) )
Theorem	rspcdv 2763*	Restricted specialization, using implicit substitution. (Contributed by NM, 17-Feb-2007.) (Revised by Mario Carneiro, 4-Jan-2017.)
|- ( ph -> A e. B )   &    |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( A. x e. B ps -> ch ) )
Theorem	rspcedv 2764*	Restricted existential specialization, using implicit substitution. (Contributed by FL, 17-Apr-2007.) (Revised by Mario Carneiro, 4-Jan-2017.)
|- ( ph -> A e. B )   &    |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( ch -> E. x e. B ps ) )
Theorem	rspcdva 2765*	Restricted specialization, using implicit substitution. (Contributed by Thierry Arnoux, 21-Jun-2020.)
|- ( x = C -> ( ps <-> ch ) )   &    |- ( ph -> A. x e. A ps )   &    |- ( ph -> C e. A )   =>    |- ( ph -> ch )
Theorem	rspcedvd 2766*	Restricted existential specialization, using implicit substitution. Variant of rspcedv 2764. (Contributed by AV, 27-Nov-2019.)
|- ( ph -> A e. B )   &    |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )   &    |- ( ph -> ch )   =>    |- ( ph -> E. x e. B ps )
Theorem	rspceaimv 2767*	Restricted existential specialization of a universally quantified implication. (Contributed by BJ, 24-Aug-2022.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( ( A e. B /\ A. y e. C ( ps -> ch ) ) -> E. x e. B A. y e. C ( ph -> ch ) )
Theorem	rspcedeq1vd 2768*	Restricted existential specialization, using implicit substitution. Variant of rspcedvd 2766 for equations, in which the left hand side depends on the quantified variable. (Contributed by AV, 24-Dec-2019.)
|- ( ph -> A e. B )   &    |- ( ( ph /\ x = A ) -> C = D )   =>    |- ( ph -> E. x e. B C = D )
Theorem	rspcedeq2vd 2769*	Restricted existential specialization, using implicit substitution. Variant of rspcedvd 2766 for equations, in which the right hand side depends on the quantified variable. (Contributed by AV, 24-Dec-2019.)
|- ( ph -> A e. B )   &    |- ( ( ph /\ x = A ) -> C = D )   =>    |- ( ph -> E. x e. B C = D )
Theorem	rspc2 2770*	2-variable restricted specialization, using implicit substitution. (Contributed by NM, 9-Nov-2012.)
|- F/ x ch   &    |- F/ y ps   &    |- ( x = A -> ( ph <-> ch ) )   &    |- ( y = B -> ( ch <-> ps ) )   =>    |- ( ( A e. C /\ B e. D ) -> ( A. x e. C A. y e. D ph -> ps ) )
Theorem	rspc2gv 2771*	Restricted specialization with two quantifiers, using implicit substitution. (Contributed by BJ, 2-Dec-2021.)
|- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )   =>    |- ( ( A e. V /\ B e. W ) -> ( A. x e. V A. y e. W ph -> ps ) )
Theorem	rspc2v 2772*	2-variable restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-1999.)
|- ( x = A -> ( ph <-> ch ) )   &    |- ( y = B -> ( ch <-> ps ) )   =>    |- ( ( A e. C /\ B e. D ) -> ( A. x e. C A. y e. D ph -> ps ) )
Theorem	rspc2va 2773*	2-variable restricted specialization, using implicit substitution. (Contributed by NM, 18-Jun-2014.)
|- ( x = A -> ( ph <-> ch ) )   &    |- ( y = B -> ( ch <-> ps ) )   =>    |- ( ( ( A e. C /\ B e. D ) /\ A. x e. C A. y e. D ph ) -> ps )
Theorem	rspc2ev 2774*	2-variable restricted existential specialization, using implicit substitution. (Contributed by NM, 16-Oct-1999.)
|- ( x = A -> ( ph <-> ch ) )   &    |- ( y = B -> ( ch <-> ps ) )   =>    |- ( ( A e. C /\ B e. D /\ ps ) -> E. x e. C E. y e. D ph )
Theorem	rspc3v 2775*	3-variable restricted specialization, using implicit substitution. (Contributed by NM, 10-May-2005.)
|- ( x = A -> ( ph <-> ch ) )   &    |- ( y = B -> ( ch <-> th ) )   &    |- ( z = C -> ( th <-> ps ) )   =>    |- ( ( A e. R /\ B e. S /\ C e. T ) -> ( A. x e. R A. y e. S A. z e. T ph -> ps ) )
Theorem	rspc3ev 2776*	3-variable restricted existentional specialization, using implicit substitution. (Contributed by NM, 25-Jul-2012.)
|- ( x = A -> ( ph <-> ch ) )   &    |- ( y = B -> ( ch <-> th ) )   &    |- ( z = C -> ( th <-> ps ) )   =>    |- ( ( ( A e. R /\ B e. S /\ C e. T ) /\ ps ) -> E. x e. R E. y e. S E. z e. T ph )
Theorem	rspceeqv 2777*	Restricted existential specialization in an equality, using implicit substitution. (Contributed by BJ, 2-Sep-2022.)
|- ( x = A -> C = D )   =>    |- ( ( A e. B /\ E = D ) -> E. x e. B E = C )
Theorem	eqvinc 2778*	A variable introduction law for class equality. (Contributed by NM, 14-Apr-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
|- A e. _V   =>    |- ( A = B <-> E. x ( x = A /\ x = B ) )
Theorem	eqvincg 2779*	A variable introduction law for class equality, deduction version. (Contributed by Thierry Arnoux, 2-Mar-2017.)
|- ( A e. V -> ( A = B <-> E. x ( x = A /\ x = B ) ) )
Theorem	eqvincf 2780	A variable introduction law for class equality, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Sep-2003.)
|- F/_ x A   &    |- F/_ x B   &    |- A e. _V   =>    |- ( A = B <-> E. x ( x = A /\ x = B ) )
Theorem	alexeq 2781*	Two ways to express substitution of A for x in ph. (Contributed by NM, 2-Mar-1995.)
|- A e. _V   =>    |- ( A. x ( x = A -> ph ) <-> E. x ( x = A /\ ph ) )
Theorem	ceqex 2782*	Equality implies equivalence with substitution. (Contributed by NM, 2-Mar-1995.)
|- ( x = A -> ( ph <-> E. x ( x = A /\ ph ) ) )
Theorem	ceqsexg 2783*	A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 11-Oct-2004.)
|- F/ x ps   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. V -> ( E. x ( x = A /\ ph ) <-> ps ) )
Theorem	ceqsexgv 2784*	Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 29-Dec-1996.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. V -> ( E. x ( x = A /\ ph ) <-> ps ) )
Theorem	ceqsrexv 2785*	Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 30-Apr-2004.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. B -> ( E. x e. B ( x = A /\ ph ) <-> ps ) )
Theorem	ceqsrexbv 2786*	Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by Mario Carneiro, 14-Mar-2014.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( E. x e. B ( x = A /\ ph ) <-> ( A e. B /\ ps ) )
Theorem	ceqsrex2v 2787*	Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 29-Oct-2005.)
|- ( x = A -> ( ph <-> ps ) )   &    |- ( y = B -> ( ps <-> ch ) )   =>    |- ( ( A e. C /\ B e. D ) -> ( E. x e. C E. y e. D ( ( x = A /\ y = B ) /\ ph ) <-> ch ) )
Theorem	clel2 2788*	An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)
|- A e. _V   =>    |- ( A e. B <-> A. x ( x = A -> x e. B ) )
Theorem	clel3g 2789*	An alternate definition of class membership when the class is a set. (Contributed by NM, 13-Aug-2005.)
|- ( B e. V -> ( A e. B <-> E. x ( x = B /\ A e. x ) ) )
Theorem	clel3 2790*	An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)
|- B e. _V   =>    |- ( A e. B <-> E. x ( x = B /\ A e. x ) )
Theorem	clel4 2791*	An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)
|- B e. _V   =>    |- ( A e. B <-> A. x ( x = B -> A e. x ) )
Theorem	pm13.183 2792*	Compare theorem *13.183 in [WhiteheadRussell] p. 178. Only A is required to be a set. (Contributed by Andrew Salmon, 3-Jun-2011.)
|- ( A e. V -> ( A = B <-> A. z ( z = A <-> z = B ) ) )
Theorem	rr19.3v 2793*	Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89. (Contributed by NM, 25-Oct-2012.)
|- ( A. x e. A A. y e. A ph <-> A. x e. A ph )
Theorem	rr19.28v 2794*	Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 29-Oct-2012.)
|- ( A. x e. A A. y e. A ( ph /\ ps ) <-> A. x e. A ( ph /\ A. y e. A ps ) )
Theorem	elabgt 2795*	Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 2799.) (Contributed by NM, 7-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
|- ( ( A e. B /\ A. x ( x = A -> ( ph <-> ps ) ) ) -> ( A e. { x | ph } <-> ps ) )
Theorem	elabgf 2796	Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. This version has bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.) (Revised by Mario Carneiro, 12-Oct-2016.)
|- F/_ x A   &    |- F/ x ps   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. B -> ( A e. { x | ph } <-> ps ) )
Theorem	elabf 2797*	Membership in a class abstraction, using implicit substitution. (Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro, 12-Oct-2016.)
|- F/ x ps   &    |- A e. _V   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. { x | ph } <-> ps )
Theorem	elab 2798*	Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 1-Aug-1994.)
|- A e. _V   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. { x | ph } <-> ps )
Theorem	elabg 2799*	Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 14-Apr-1995.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. V -> ( A e. { x | ph } <-> ps ) )
Theorem	elab2g 2800*	Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)
|- ( x = A -> ( ph <-> ps ) )   &    |- B = { x | ph }   =>    |- ( A e. V -> ( A e. B <-> ps ) )