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	cbvreu 2701*	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 2702*	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 2703*	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 2704*	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 2705*	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 2706*	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 2707*	Version of cbvralv 2703 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 )
Theorem	cbvrexvw 2708*	Version of cbvrexv 2704 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 ) )   =>    |- ( E. x e. A ph <-> E. y e. A ps )
Theorem	cbvreuvw 2709*	Version of cbvreuv 2705 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 ) )   =>    |- ( E! x e. A ph <-> E! y e. A ps )
Theorem	cbvraldva2 2710*	Rule used to change the bound variable in a restricted universal quantifier with implicit substitution which also changes the quantifier domain. Deduction form. (Contributed by David Moews, 1-May-2017.)
|- ( ( ph /\ x = y ) -> ( ps <-> ch ) )   &    |- ( ( ph /\ x = y ) -> A = B )   =>    |- ( ph -> ( A. x e. A ps <-> A. y e. B ch ) )
Theorem	cbvrexdva2 2711*	Rule used to change the bound variable in a restricted existential quantifier with implicit substitution which also changes the quantifier domain. Deduction form. (Contributed by David Moews, 1-May-2017.)
|- ( ( ph /\ x = y ) -> ( ps <-> ch ) )   &    |- ( ( ph /\ x = y ) -> A = B )   =>    |- ( ph -> ( E. x e. A ps <-> E. y e. B ch ) )
Theorem	cbvraldva 2712*	Rule used to change the bound variable in a restricted universal quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
|- ( ( ph /\ x = y ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( A. x e. A ps <-> A. y e. A ch ) )
Theorem	cbvrexdva 2713*	Rule used to change the bound variable in a restricted existential quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
|- ( ( ph /\ x = y ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( E. x e. A ps <-> E. y e. A ch ) )
Theorem	cbvral2vw 2714*	Change bound variables of double restricted universal quantification, using implicit substitution. Version of cbvral2v 2716 with a disjoint variable condition, which does not require ax-13 2150. (Contributed by NM, 10-Aug-2004.) (Revised by Gino Giotto, 10-Jan-2024.)
|- ( x = z -> ( ph <-> ch ) )   &    |- ( y = w -> ( ch <-> ps ) )   =>    |- ( A. x e. A A. y e. B ph <-> A. z e. A A. w e. B ps )
Theorem	cbvrex2vw 2715*	Change bound variables of double restricted universal quantification, using implicit substitution. Version of cbvrex2v 2717 with a disjoint variable condition, which does not require ax-13 2150. (Contributed by FL, 2-Jul-2012.) (Revised by Gino Giotto, 10-Jan-2024.)
|- ( x = z -> ( ph <-> ch ) )   &    |- ( y = w -> ( ch <-> ps ) )   =>    |- ( E. x e. A E. y e. B ph <-> E. z e. A E. w e. B ps )
Theorem	cbvral2v 2716*	Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by NM, 10-Aug-2004.)
|- ( x = z -> ( ph <-> ch ) )   &    |- ( y = w -> ( ch <-> ps ) )   =>    |- ( A. x e. A A. y e. B ph <-> A. z e. A A. w e. B ps )
Theorem	cbvrex2v 2717*	Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by FL, 2-Jul-2012.)
|- ( x = z -> ( ph <-> ch ) )   &    |- ( y = w -> ( ch <-> ps ) )   =>    |- ( E. x e. A E. y e. B ph <-> E. z e. A E. w e. B ps )
Theorem	cbvral3v 2718*	Change bound variables of triple restricted universal quantification, using implicit substitution. (Contributed by NM, 10-May-2005.)
|- ( x = w -> ( ph <-> ch ) )   &    |- ( y = v -> ( ch <-> th ) )   &    |- ( z = u -> ( th <-> ps ) )   =>    |- ( A. x e. A A. y e. B A. z e. C ph <-> A. w e. A A. v e. B A. u e. C ps )
Theorem	cbvralsv 2719*	Change bound variable by using a substitution. (Contributed by NM, 20-Nov-2005.) (Revised by Andrew Salmon, 11-Jul-2011.)
|- ( A. x e. A ph <-> A. y e. A [ y / x ] ph )
Theorem	cbvrexsv 2720*	Change bound variable by using a substitution. (Contributed by NM, 2-Mar-2008.) (Revised by Andrew Salmon, 11-Jul-2011.)
|- ( E. x e. A ph <-> E. y e. A [ y / x ] ph )
Theorem	sbralie 2721*	Implicit to explicit substitution that swaps variables in a quantified expression. (Contributed by NM, 5-Sep-2004.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( A. x e. y ph <-> [ y / x ] A. y e. x ps )
Theorem	rabbiia 2722	Equivalent wff's yield equal restricted class abstractions (inference form). (Contributed by NM, 22-May-1999.)
|- ( x e. A -> ( ph <-> ps ) )   =>    |- { x e. A | ph } = { x e. A | ps }
Theorem	rabbii 2723	Equivalent wff's correspond to equal restricted class abstractions. Inference form of rabbidv 2726. (Contributed by Peter Mazsa, 1-Nov-2019.)
|- ( ph <-> ps )   =>    |- { x e. A | ph } = { x e. A | ps }
Theorem	rabbidva2 2724*	Equivalent wff's yield equal restricted class abstractions. (Contributed by Thierry Arnoux, 4-Feb-2017.)
|- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. B /\ ch ) ) )   =>    |- ( ph -> { x e. A | ps } = { x e. B | ch } )
Theorem	rabbidva 2725*	Equivalent wff's yield equal restricted class abstractions (deduction form). (Contributed by NM, 28-Nov-2003.)
|- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )   =>    |- ( ph -> { x e. A | ps } = { x e. A | ch } )
Theorem	rabbidv 2726*	Equivalent wff's yield equal restricted class abstractions (deduction form). (Contributed by NM, 10-Feb-1995.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> { x e. A | ps } = { x e. A | ch } )
Theorem	rabeqf 2727	Equality theorem for restricted class abstractions, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.)
|- F/_ x A   &    |- F/_ x B   =>    |- ( A = B -> { x e. A | ph } = { x e. B | ph } )
Theorem	rabeqif 2728	Equality theorem for restricted class abstractions. Inference form of rabeqf 2727. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
|- F/_ x A   &    |- F/_ x B   &    |- A = B   =>    |- { x e. A | ph } = { x e. B | ph }
Theorem	rabeq 2729*	Equality theorem for restricted class abstractions. (Contributed by NM, 15-Oct-2003.)
|- ( A = B -> { x e. A | ph } = { x e. B | ph } )
Theorem	rabeqi 2730*	Equality theorem for restricted class abstractions. Inference form of rabeq 2729. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
|- A = B   =>    |- { x e. A | ph } = { x e. B | ph }
Theorem	rabeqdv 2731*	Equality of restricted class abstractions. Deduction form of rabeq 2729. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- ( ph -> A = B )   =>    |- ( ph -> { x e. A | ps } = { x e. B | ps } )
Theorem	rabeqbidv 2732*	Equality of restricted class abstractions. (Contributed by Jeff Madsen, 1-Dec-2009.)
|- ( ph -> A = B )   &    |- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> { x e. A | ps } = { x e. B | ch } )
Theorem	rabeqbidva 2733*	Equality of restricted class abstractions. (Contributed by Mario Carneiro, 26-Jan-2017.)
|- ( ph -> A = B )   &    |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )   =>    |- ( ph -> { x e. A | ps } = { x e. B | ch } )
Theorem	rabeq2i 2734	Inference from equality of a class variable and a restricted class abstraction. (Contributed by NM, 16-Feb-2004.)
|- A = { x e. B | ph }   =>    |- ( x e. A <-> ( x e. B /\ ph ) )
Theorem	cbvrab 2735	Rule to change the bound variable in a restricted class abstraction, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 9-Oct-2016.)
|- F/_ x A   &    |- F/_ y A   &    |- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- { x e. A | ph } = { y e. A | ps }
Theorem	cbvrabv 2736*	Rule to change the bound variable in a restricted class abstraction, using implicit substitution. (Contributed by NM, 26-May-1999.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- { x e. A | ph } = { y e. A | ps }
2.1.6  The universal class
Syntax	cvv 2737	Extend class notation to include the universal class symbol.
class _V
Theorem	vjust 2738	Soundness justification theorem for df-v 2739. (Contributed by Rodolfo Medina, 27-Apr-2010.)
|- { x | x = x } = { y | y = y }
Definition	df-v 2739	Define the universal class. Definition 5.20 of [TakeutiZaring] p. 21. Also Definition 2.9 of [Quine] p. 19. (Contributed by NM, 5-Aug-1993.)
|- _V = { x | x = x }
Theorem	vex 2740	All setvar variables are sets (see isset 2743). Theorem 6.8 of [Quine] p. 43. (Contributed by NM, 5-Aug-1993.)
|- x e. _V
Theorem	elv 2741	Technical lemma used to shorten proofs. If a proposition is implied by x e. _V (which is true, see vex 2740), then it is true. (Contributed by Peter Mazsa, 13-Oct-2018.)
|- ( x e. _V -> ph )   =>    |- ph
Theorem	elvd 2742	Technical lemma used to shorten proofs. If a proposition is implied by x e. _V (which is true, see vex 2740) and another antecedent, then it is implied by the other antecedent. (Contributed by Peter Mazsa, 23-Oct-2018.)
|- ( ( ph /\ x e. _V ) -> ps )   =>    |- ( ph -> ps )
Theorem	isset 2743*	Two ways to say " A is a set": A class A is a member of the universal class _V (see df-v 2739) if and only if the class A exists (i.e. there exists some set x equal to class A). Theorem 6.9 of [Quine] p. 43. Notational convention: We will use the notational device " A e. _V " to mean " A is a set" very frequently, for example in uniex 4434. Note the when A is not a set, it is called a proper class. In some theorems, such as uniexg 4436, in order to shorten certain proofs we use the more general antecedent A e. V instead of A e. _V to mean " A is a set." Note that a constant is implicitly considered distinct from all variables. This is why _V is not included in the distinct variable list, even though df-clel 2173 requires that the expression substituted for B not contain x. (Also, the Metamath spec does not allow constants in the distinct variable list.) (Contributed by NM, 26-May-1993.)
|- ( A e. _V <-> E. x x = A )
Theorem	issetf 2744	A version of isset that does not require x and A to be distinct. (Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro, 10-Oct-2016.)
|- F/_ x A   =>    |- ( A e. _V <-> E. x x = A )
Theorem	isseti 2745*	A way to say " A is a set" (inference form). (Contributed by NM, 5-Aug-1993.)
|- A e. _V   =>    |- E. x x = A
Theorem	issetri 2746*	A way to say " A is a set" (inference form). (Contributed by NM, 5-Aug-1993.)
|- E. x x = A   =>    |- A e. _V
Theorem	eqvisset 2747	A class equal to a variable is a set. Note the absence of disjoint variable condition, contrary to isset 2743 and issetri 2746. (Contributed by BJ, 27-Apr-2019.)
|- ( x = A -> A e. _V )
Theorem	elex 2748	If a class is a member of another class, then it is a set. Theorem 6.12 of [Quine] p. 44. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
|- ( A e. B -> A e. _V )
Theorem	elexi 2749	If a class is a member of another class, it is a set. (Contributed by NM, 11-Jun-1994.)
|- A e. B   =>    |- A e. _V
Theorem	elexd 2750	If a class is a member of another class, it is a set. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ( ph -> A e. V )   =>    |- ( ph -> A e. _V )
Theorem	elisset 2751*	An element of a class exists. (Contributed by NM, 1-May-1995.)
|- ( A e. V -> E. x x = A )
Theorem	elex22 2752*	If two classes each contain another class, then both contain some set. (Contributed by Alan Sare, 24-Oct-2011.)
|- ( ( A e. B /\ A e. C ) -> E. x ( x e. B /\ x e. C ) )
Theorem	elex2 2753*	If a class contains another class, then it contains some set. (Contributed by Alan Sare, 25-Sep-2011.)
|- ( A e. B -> E. x x e. B )
Theorem	ralv 2754	A universal quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)
|- ( A. x e. _V ph <-> A. x ph )
Theorem	rexv 2755	An existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)
|- ( E. x e. _V ph <-> E. x ph )
Theorem	reuv 2756	A unique existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 1-Nov-2010.)
|- ( E! x e. _V ph <-> E! x ph )
Theorem	rmov 2757	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 2758	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 2759*	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 2760*	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 2761*	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 2762*	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 2763*	Closed theorem version of ceqsalg 2765. (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 2764*	Restricted quantifier version of ceqsalt 2763. (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 2765*	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 2766*	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 2767*	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 2768*	Restricted quantifier version of ceqsalv 2767. (Contributed by NM, 21-Jun-2013.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. B -> ( A. x e. B ( x = A -> ph ) <-> ps ) )
Theorem	gencl 2769*	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 2770*	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 2771*	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 2772*	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 2773*	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 2774*	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 2775*	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 2776*	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 2777*	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 2778*	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 2779*	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 2780*	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 2781*	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 2782*	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 2783*	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 2784*	Restatement of gencbvex 2783 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 2785*	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 2786*	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 2787	Closed theorem form of vtoclgf 2795. (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 2788	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 2789*	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 2790*	Implicit substitution of a class for a setvar variable. This is a generalization of chvar 1757. (Contributed by NM, 30-Aug-1993.)
|- F/ x ps   &    |- A e. _V   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ph   =>    |- ps
Theorem	vtocl 2791*	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 2792*	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 2793*	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 2794*	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 2795	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 2796*	Version of vtoclgf 2795 with one nonfreeness hypothesis replaced with a disjoint variable condition, thus avoiding dependency on ax-11 1506 and ax-13 2150. (Contributed by BJ, 1-May-2019.)
|- F/ x ps   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ph   =>    |- ( A e. V -> ps )
Theorem	vtoclg 2797*	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 2798*	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 2799	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 2800	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 )