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	gencbvex2 2601*	Restatement of gencbvex 2600 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 2602*	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 2603*	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 2604	Closed theorem form of vtoclgf 2612. (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 2605	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 2606*	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 2607*	Implicit substitution of a class for a setvar variable. This is a generalization of chvar 1640. (Contributed by NM, 30-Aug-1993.)
|- F/ x ps   &    |- A e. _V   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ph   =>    |- ps
Theorem	vtocl 2608*	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 2609*	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 2610*	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 2611*	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 2612	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	vtoclg 2613*	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 2614*	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 2615	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 2616	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 2617*	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 2618*	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 2619*	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 2620*	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 2621*	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 2622*	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 2623*	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 2624*	Implicit substitution of a class for a setvar variable. (Contributed by NM, 10-Jan-2004.)
|- ( x = A -> ph )   =>    |- ( A e. V -> ph )
Theorem	vtoclegft 2625*	Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef 2626.) (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 2626*	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 2627*	Implicit substitution of a class for a setvar variable. (Contributed by NM, 9-Sep-1993.)
|- A e. _V   &    |- ( x = A -> ph )   =>    |- ph
Theorem	vtoclri 2628*	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 2629	A closed version of spcimgf 2633. (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 2630	A closed version of spcgf 2635. (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 2631	A closed version of spcimegf 2634. (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 2632	A closed version of spcegf 2636. (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 2633	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 2634	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 2635	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 2636	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 2637*	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 2638*	Rule of specialization, using implicit substitution. Analogous to rspcdv 2659. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. B )   &    |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( A. x ps -> ch ) )
Theorem	spcimedv 2639*	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 2640*	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 2641*	Existential specialization, using implicit substitution. (Contributed by NM, 14-Aug-1994.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. V -> ( ps -> E. x ph ) )
Theorem	spc2egv 2642*	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 2643*	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 2644*	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 2645*	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 2646*	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 2647*	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 2648*	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 2649*	A closed version of rspc 2650. (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 2650*	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 2651*	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 2652*	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 2653*	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 2654*	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 2655*	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 2656*	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 2657*	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 2658*	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 2659*	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 2660*	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	rspc2 2661*	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	rspc2v 2662*	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 2663*	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 2664*	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 2665*	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 2666*	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	eqvinc 2667*	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 2668*	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 2669	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 2670*	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 2671*	Equality implies equivalence with substitution. (Contributed by NM, 2-Mar-1995.)
|- ( x = A -> ( ph <-> E. x ( x = A /\ ph ) ) )
Theorem	ceqsexg 2672*	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 2673*	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 2674*	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 2675*	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 2676*	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 2677*	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 2678*	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 2679*	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 2680*	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 2681*	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 2682*	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 2683*	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 2684*	Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 2688.) (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 2685	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 2686*	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 2687*	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 2688*	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 2689*	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 ) )
Theorem	elab2 2690*	Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)
|- A e. _V   &    |- ( x = A -> ( ph <-> ps ) )   &    |- B = { x | ph }   =>    |- ( A e. B <-> ps )
Theorem	elab4g 2691*	Membership in a class abstraction, using implicit substitution. (Contributed by NM, 17-Oct-2012.)
|- ( x = A -> ( ph <-> ps ) )   &    |- B = { x | ph }   =>    |- ( A e. B <-> ( A e. _V /\ ps ) )
Theorem	elab3gf 2692	Membership in a class abstraction, with a weaker antecedent than elabgf 2685. (Contributed by NM, 6-Sep-2011.)
|- F/_ x A   &    |- F/ x ps   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( ( ps -> A e. B ) -> ( A e. { x | ph } <-> ps ) )
Theorem	elab3g 2693*	Membership in a class abstraction, with a weaker antecedent than elabg 2688. (Contributed by NM, 29-Aug-2006.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( ( ps -> A e. B ) -> ( A e. { x | ph } <-> ps ) )
Theorem	elab3 2694*	Membership in a class abstraction using implicit substitution. (Contributed by NM, 10-Nov-2000.)
|- ( ps -> A e. _V )   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. { x | ph } <-> ps )
Theorem	elrabi 2695*	Implication for the membership in a restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.)
|- ( A e. { x e. V | ph } -> A e. V )
Theorem	elrabf 2696	Membership in a restricted class abstraction, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.)
|- F/_ x A   &    |- F/_ x B   &    |- F/ x ps   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. { x e. B | ph } <-> ( A e. B /\ ps ) )
Theorem	elrab3t 2697*	Membership in a restricted class abstraction, using implicit substitution. (Closed theorem version of elrab3 2699.) (Contributed by Thierry Arnoux, 31-Aug-2017.)
|- ( ( A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B ) -> ( A e. { x e. B | ph } <-> ps ) )
Theorem	elrab 2698*	Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 21-May-1999.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. { x e. B | ph } <-> ( A e. B /\ ps ) )
Theorem	elrab3 2699*	Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 5-Oct-2006.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. B -> ( A e. { x e. B | ph } <-> ps ) )
Theorem	elrab2 2700*	Membership in a class abstraction, using implicit substitution. (Contributed by NM, 2-Nov-2006.)
|- ( x = A -> ( ph <-> ps ) )   &    |- C = { x e. B | ph }   =>    |- ( A e. C <-> ( A e. B /\ ps ) )