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	rmov 2701	An at-most-one quantifier restricted to the universe is unrestricted. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
|- ( E* x e. _V ph <-> E* x ph )
Theorem	rabab 2702	A class abstraction restricted to the universe is unrestricted. (Contributed by NM, 27-Dec-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
|- { x e. _V | ph } = { x | ph }
Theorem	ralcom4 2703*	Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
|- ( A. x e. A A. y ph <-> A. y A. x e. A ph )
Theorem	rexcom4 2704*	Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
|- ( E. x e. A E. y ph <-> E. y E. x e. A ph )
Theorem	rexcom4a 2705*	Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)
|- ( E. x E. y e. A ( ph /\ ps ) <-> E. y e. A ( ph /\ E. x ps ) )
Theorem	rexcom4b 2706*	Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)
|- B e. _V   =>    |- ( E. x E. y e. A ( ph /\ x = B ) <-> E. y e. A ph )
Theorem	ceqsalt 2707*	Closed theorem version of ceqsalg 2709. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
|- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. V ) -> ( A. x ( x = A -> ph ) <-> ps ) )
Theorem	ceqsralt 2708*	Restricted quantifier version of ceqsalt 2707. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
|- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B ) -> ( A. x e. B ( x = A -> ph ) <-> ps ) )
Theorem	ceqsalg 2709*	A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 29-Oct-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
|- F/ x ps   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. V -> ( A. x ( x = A -> ph ) <-> ps ) )
Theorem	ceqsal 2710*	A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.)
|- F/ x ps   &    |- A e. _V   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( A. x ( x = A -> ph ) <-> ps )
Theorem	ceqsalv 2711*	A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.)
|- A e. _V   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( A. x ( x = A -> ph ) <-> ps )
Theorem	ceqsralv 2712*	Restricted quantifier version of ceqsalv 2711. (Contributed by NM, 21-Jun-2013.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. B -> ( A. x e. B ( x = A -> ph ) <-> ps ) )
Theorem	gencl 2713*	Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
|- ( th <-> E. x ( ch /\ A = B ) )   &    |- ( A = B -> ( ph <-> ps ) )   &    |- ( ch -> ph )   =>    |- ( th -> ps )
Theorem	2gencl 2714*	Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
|- ( C e. S <-> E. x e. R A = C )   &    |- ( D e. S <-> E. y e. R B = D )   &    |- ( A = C -> ( ph <-> ps ) )   &    |- ( B = D -> ( ps <-> ch ) )   &    |- ( ( x e. R /\ y e. R ) -> ph )   =>    |- ( ( C e. S /\ D e. S ) -> ch )
Theorem	3gencl 2715*	Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
|- ( D e. S <-> E. x e. R A = D )   &    |- ( F e. S <-> E. y e. R B = F )   &    |- ( G e. S <-> E. z e. R C = G )   &    |- ( A = D -> ( ph <-> ps ) )   &    |- ( B = F -> ( ps <-> ch ) )   &    |- ( C = G -> ( ch <-> th ) )   &    |- ( ( x e. R /\ y e. R /\ z e. R ) -> ph )   =>    |- ( ( D e. S /\ F e. S /\ G e. S ) -> th )
Theorem	cgsexg 2716*	Implicit substitution inference for general classes. (Contributed by NM, 26-Aug-2007.)
|- ( x = A -> ch )   &    |- ( ch -> ( ph <-> ps ) )   =>    |- ( A e. V -> ( E. x ( ch /\ ph ) <-> ps ) )
Theorem	cgsex2g 2717*	Implicit substitution inference for general classes. (Contributed by NM, 26-Jul-1995.)
|- ( ( x = A /\ y = B ) -> ch )   &    |- ( ch -> ( ph <-> ps ) )   =>    |- ( ( A e. V /\ B e. W ) -> ( E. x E. y ( ch /\ ph ) <-> ps ) )
Theorem	cgsex4g 2718*	An implicit substitution inference for 4 general classes. (Contributed by NM, 5-Aug-1995.)
|- ( ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) -> ch )   &    |- ( ch -> ( ph <-> ps ) )   =>    |- ( ( ( A e. R /\ B e. S ) /\ ( C e. R /\ D e. S ) ) -> ( E. x E. y E. z E. w ( ch /\ ph ) <-> ps ) )
Theorem	ceqsex 2719*	Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) (Revised by Mario Carneiro, 10-Oct-2016.)
|- F/ x ps   &    |- A e. _V   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( E. x ( x = A /\ ph ) <-> ps )
Theorem	ceqsexv 2720*	Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.)
|- A e. _V   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( E. x ( x = A /\ ph ) <-> ps )
Theorem	ceqsex2 2721*	Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.)
|- F/ x ps   &    |- F/ y ch   &    |- A e. _V   &    |- B e. _V   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ( y = B -> ( ps <-> ch ) )   =>    |- ( E. x E. y ( x = A /\ y = B /\ ph ) <-> ch )
Theorem	ceqsex2v 2722*	Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.)
|- A e. _V   &    |- B e. _V   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ( y = B -> ( ps <-> ch ) )   =>    |- ( E. x E. y ( x = A /\ y = B /\ ph ) <-> ch )
Theorem	ceqsex3v 2723*	Elimination of three existential quantifiers, using implicit substitution. (Contributed by NM, 16-Aug-2011.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ( y = B -> ( ps <-> ch ) )   &    |- ( z = C -> ( ch <-> th ) )   =>    |- ( E. x E. y E. z ( ( x = A /\ y = B /\ z = C ) /\ ph ) <-> th )
Theorem	ceqsex4v 2724*	Elimination of four existential quantifiers, using implicit substitution. (Contributed by NM, 23-Sep-2011.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- D e. _V   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ( y = B -> ( ps <-> ch ) )   &    |- ( z = C -> ( ch <-> th ) )   &    |- ( w = D -> ( th <-> ta ) )   =>    |- ( E. x E. y E. z E. w ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) /\ ph ) <-> ta )
Theorem	ceqsex6v 2725*	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 2726*	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 2727*	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 2728*	Restatement of gencbvex 2727 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 2729*	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 2730*	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 2731	Closed theorem form of vtoclgf 2739. (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 2732	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 2733*	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 2734*	Implicit substitution of a class for a setvar variable. This is a generalization of chvar 1730. (Contributed by NM, 30-Aug-1993.)
|- F/ x ps   &    |- A e. _V   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ph   =>    |- ps
Theorem	vtocl 2735*	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 2736*	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 2737*	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 2738*	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 2739	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 2740*	Version of vtoclgf 2739 with one non-freeness hypothesis replaced with a disjoint variable condition, thus avoiding dependency on ax-11 1484 and ax-13 1491. (Contributed by BJ, 1-May-2019.)
|- F/ x ps   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ph   =>    |- ( A e. V -> ps )
Theorem	vtoclg 2741*	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 2742*	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 2743	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 2744	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 2745*	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 2746*	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 2747*	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 2748*	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 2749*	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 2750*	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 2751*	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 2752*	Implicit substitution of a class for a setvar variable. (Contributed by NM, 10-Jan-2004.)
|- ( x = A -> ph )   =>    |- ( A e. V -> ph )
Theorem	vtoclegft 2753*	Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef 2754.) (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 2754*	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 2755*	Implicit substitution of a class for a setvar variable. (Contributed by NM, 9-Sep-1993.)
|- A e. _V   &    |- ( x = A -> ph )   =>    |- ph
Theorem	vtoclri 2756*	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 2757	A closed version of spcimgf 2761. (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 2758	A closed version of spcgf 2763. (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 2759	A closed version of spcimegf 2762. (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 2760	A closed version of spcegf 2764. (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 2761	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 2762	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 2763	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 2764	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 2765*	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 2766*	Rule of specialization, using implicit substitution. Analogous to rspcdv 2787. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. B )   &    |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( A. x ps -> ch ) )
Theorem	spcimedv 2767*	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 2768*	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 2769*	Existential specialization, using implicit substitution. (Contributed by NM, 14-Aug-1994.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. V -> ( ps -> E. x ph ) )
Theorem	spc2egv 2770*	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 2771*	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 2772*	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 2773*	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 2774*	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 2775*	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 2776*	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 2777*	A closed version of rspc 2778. (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 2778*	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 2779*	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 2780*	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 2781*	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 2782*	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 2783*	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 2784*	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 2785*	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 2786*	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 2787*	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 2788*	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 2789*	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 2790*	Restricted existential specialization, using implicit substitution. Variant of rspcedv 2788. (Contributed by AV, 27-Nov-2019.)
|- ( ph -> A e. B )   &    |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )   &    |- ( ph -> ch )   =>    |- ( ph -> E. x e. B ps )
Theorem	rspcime 2791*	Prove a restricted existential. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- ( ( ph /\ x = A ) -> ps )   &    |- ( ph -> A e. B )   =>    |- ( ph -> E. x e. B ps )
Theorem	rspceaimv 2792*	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 2793*	Restricted existential specialization, using implicit substitution. Variant of rspcedvd 2790 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 2794*	Restricted existential specialization, using implicit substitution. Variant of rspcedvd 2790 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 2795*	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 2796*	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 2797*	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 2798*	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 2799*	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 2800*	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 ) )