P. 26 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 2501-2600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	reueq1 2501*	Equality theorem for restricted uniqueness quantifier. (Contributed by NM, 5-Apr-2004.)
|- (A = B -> (E!x e. A ph <-> E!x e. B ph))
Theorem	rmoeq1 2502*	Equality theorem for restricted uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
|- (A = B -> (E*x e. A ph <-> E*x e. B ph))
Theorem	raleqi 2503*	Equality inference for restricted universal qualifier. (Contributed by Paul Chapman, 22-Jun-2011.)
|- A = B   =>   |- (A.x e. A ph <-> A.x e. B ph)
Theorem	rexeqi 2504*	Equality inference for restricted existential qualifier. (Contributed by Mario Carneiro, 23-Apr-2015.)
|- A = B   =>   |- (E.x e. A ph <-> E.x e. B ph)
Theorem	raleqdv 2505*	Equality deduction for restricted universal quantifier. (Contributed by NM, 13-Nov-2005.)
|- (ph -> A = B)   =>   |- (ph -> (A.x e. A ps <-> A.x e. B ps))
Theorem	rexeqdv 2506*	Equality deduction for restricted existential quantifier. (Contributed by NM, 14-Jan-2007.)
|- (ph -> A = B)   =>   |- (ph -> (E.x e. A ps <-> E.x e. B ps))
Theorem	raleqbi1dv 2507*	Equality deduction for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.)
|- (A = B -> (ph <-> ps))   =>   |- (A = B -> (A.x e. A ph <-> A.x e. B ps))
Theorem	rexeqbi1dv 2508*	Equality deduction for restricted existential quantifier. (Contributed by NM, 18-Mar-1997.)
|- (A = B -> (ph <-> ps))   =>   |- (A = B -> (E.x e. A ph <-> E.x e. B ps))
Theorem	reueqd 2509*	Equality deduction for restricted uniqueness quantifier. (Contributed by NM, 5-Apr-2004.)
|- (A = B -> (ph <-> ps))   =>   |- (A = B -> (E!x e. A ph <-> E!x e. B ps))
Theorem	rmoeqd 2510*	Equality deduction for restricted uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
|- (A = B -> (ph <-> ps))   =>   |- (A = B -> (E*x e. A ph <-> E*x e. B ps))
Theorem	raleqbidv 2511*	Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.)
|- (ph -> A = B)   &   |- (ph -> (ps <-> ch))   =>   |- (ph -> (A.x e. A ps <-> A.x e. B ch))
Theorem	rexeqbidv 2512*	Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.)
|- (ph -> A = B)   &   |- (ph -> (ps <-> ch))   =>   |- (ph -> (E.x e. A ps <-> E.x e. B ch))
Theorem	raleqbidva 2513*	Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)
|- (ph -> A = B)   &   |- ((ph /\ x e. A) -> (ps <-> ch))   =>   |- (ph -> (A.x e. A ps <-> A.x e. B ch))
Theorem	rexeqbidva 2514*	Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)
|- (ph -> A = B)   &   |- ((ph /\ x e. A) -> (ps <-> ch))   =>   |- (ph -> (E.x e. A ps <-> E.x e. B ch))
Theorem	mormo 2515	Unrestricted "at most one" implies restricted "at most one". (Contributed by NM, 16-Jun-2017.)
|- (E*xph -> E*x e. A ph)
Theorem	reu5 2516	Restricted uniqueness in terms of "at most one." (Contributed by NM, 23-May-1999.) (Revised by NM, 16-Jun-2017.)
|- (E!x e. A ph <-> (E.x e. A ph /\ E*x e. A ph))
Theorem	reurex 2517	Restricted unique existence implies restricted existence. (Contributed by NM, 19-Aug-1999.)
|- (E!x e. A ph -> E.x e. A ph)
Theorem	reurmo 2518	Restricted existential uniqueness implies restricted "at most one." (Contributed by NM, 16-Jun-2017.)
|- (E!x e. A ph -> E*x e. A ph)
Theorem	rmo5 2519	Restricted "at most one" in term of uniqueness. (Contributed by NM, 16-Jun-2017.)
|- (E*x e. A ph <-> (E.x e. A ph -> E!x e. A ph))
Theorem	nrexrmo 2520	Nonexistence implies restricted "at most one". (Contributed by NM, 17-Jun-2017.)
|- (-. E.x e. A ph -> E*x e. A ph)
Theorem	cbvralf 2521	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 7-Mar-2004.) (Revised by Mario Carneiro, 9-Oct-2016.)
|- F/_xA   &   |- F/_yA   &   |- F/yph   &   |- F/xps   &   |- (x = y -> (ph <-> ps))   =>   |- (A.x e. A ph <-> A.y e. A ps)
Theorem	cbvrexf 2522	Rule used to change bound variables, using implicit substitution. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 9-Oct-2016.) (Proof rewritten by Jim Kingdon, 10-Jun-2018.)
|- F/_xA   &   |- F/_yA   &   |- F/yph   &   |- F/xps   &   |- (x = y -> (ph <-> ps))   =>   |- (E.x e. A ph <-> E.y e. A ps)
Theorem	cbvral 2523*	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.)
|- F/yph   &   |- F/xps   &   |- (x = y -> (ph <-> ps))   =>   |- (A.x e. A ph <-> A.y e. A ps)
Theorem	cbvrex 2524*	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
|- F/yph   &   |- F/xps   &   |- (x = y -> (ph <-> ps))   =>   |- (E.x e. A ph <-> E.y e. A ps)
Theorem	cbvreu 2525*	Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by Mario Carneiro, 15-Oct-2016.)
|- F/yph   &   |- F/xps   &   |- (x = y -> (ph <-> ps))   =>   |- (E!x e. A ph <-> E!y e. A ps)
Theorem	cbvrmo 2526*	Change the bound variable of restricted "at most one" using implicit substitution. (Contributed by NM, 16-Jun-2017.)
|- F/yph   &   |- F/xps   &   |- (x = y -> (ph <-> ps))   =>   |- (E*x e. A ph <-> E*y e. A ps)
Theorem	cbvralv 2527*	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 2528*	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 2529*	Change the bound variable of a restricted uniqueness 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 2530*	Change the bound variable of a restricted uniqueness 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	cbvraldva2 2531*	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 2532*	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 2533*	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 2534*	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	cbvral2v 2535*	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 2536*	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 2537*	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 2538*	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 2539*	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 2540*	Implicit to explicit substitution that swaps variables in a quantified expression. (Contributed by NM, 5-Sep-2004.)
|- (x = y -> (ph <-> ps))   =>   |- ([x / y]A.x e. y ph <-> A.y e. x ps)
Theorem	rabbiia 2541	Equivalent wff's yield equal restricted class abstractions (inference rule). (Contributed by NM, 22-May-1999.)
|- (x e. A -> (ph <-> ps))   =>   |- {x e. A | ph } = {x e. A | ps }
Theorem	rabbidva 2542*	Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 28-Nov-2003.)
|- ((ph /\ x e. A) -> (ps <-> ch))   =>   |- (ph -> {x e. A | ps } = {x e. A | ch })
Theorem	rabbidv 2543*	Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 10-Feb-1995.)
|- (ph -> (ps <-> ch))   =>   |- (ph -> {x e. A | ps } = {x e. A | ch })
Theorem	rabeqf 2544	Equality theorem for restricted class abstractions, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.)
|- F/_xA   &   |- F/_xB   =>   |- (A = B -> {x e. A | ph } = {x e. B | ph })
Theorem	rabeq 2545*	Equality theorem for restricted class abstractions. (Contributed by NM, 15-Oct-2003.)
|- (A = B -> {x e. A | ph } = {x e. B | ph })
Theorem	rabeqbidv 2546*	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 2547*	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 2548	Inference rule 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 2549	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/_xA   &   |- F/_yA   &   |- F/yph   &   |- F/xps   &   |- (x = y -> (ph <-> ps))   =>   |- {x e. A | ph } = {y e. A | ps }
Theorem	cbvrabv 2550*	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 2551	Extend class notation to include the universal class symbol.
class _V
Theorem	vjust 2552	Soundness justification theorem for df-v 2553. (Contributed by Rodolfo Medina, 27-Apr-2010.)
|- {x | x = x } = {y | y = y }
Definition	df-v 2553	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 2554	All setvar variables are sets (see isset 2555). Theorem 6.8 of [Quine] p. 43. (Contributed by NM, 5-Aug-1993.)
|- x e. _V
Theorem	isset 2555*	Two ways to say "A is a set": A class A is a member of the universal class _V (see df-v 2553) 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 . Note the when A is not a set, it is called a proper class. In some theorems, such as uniexg , 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 2033 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, 5-Aug-1993.)
|- (A e. _V <-> E.x x = A)
Theorem	issetf 2556	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/_xA   =>   |- (A e. _V <-> E.x x = A)
Theorem	isseti 2557*	A way to say "A is a set" (inference rule). (Contributed by NM, 5-Aug-1993.)
|- A e. _V   =>   |- E.x x = A
Theorem	issetri 2558*	A way to say "A is a set" (inference rule). (Contributed by NM, 5-Aug-1993.)
|- E.x x = A   =>   |- A e. _V
Theorem	eqvisset 2559	A class equal to a variable is a set. Note the absence of dv condition, contrary to isset 2555 and issetri 2558. (Contributed by BJ, 27-Apr-2019.)
|- (x = A -> A e. _V)
Theorem	elex 2560	If a class is a member of another class, 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 2561	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	elisset 2562*	An element of a class exists. (Contributed by NM, 1-May-1995.)
|- (A e. V -> E.x x = A)
Theorem	elex22 2563*	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 2564*	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 2565	A universal quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)
|- (A.x e. _V ph <-> A.xph)
Theorem	rexv 2566	An existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)
|- (E.x e. _V ph <-> E.xph)
Theorem	reuv 2567	A uniqueness quantifier restricted to the universe is unrestricted. (Contributed by NM, 1-Nov-2010.)
|- (E!x e. _V ph <-> E!xph)
Theorem	rmov 2568	A uniqueness quantifier restricted to the universe is unrestricted. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
|- (E*x e. _V ph <-> E*xph)
Theorem	rabab 2569	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 2570*	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.yph <-> A.yA.x e. A ph)
Theorem	rexcom4 2571*	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.yph <-> E.yE.x e. A ph)
Theorem	rexcom4a 2572*	Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)
|- (E.xE.y e. A (ph /\ ps) <-> E.y e. A (ph /\ E.xps))
Theorem	rexcom4b 2573*	Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)
|- B e. _V   =>   |- (E.xE.y e. A (ph /\ x = B) <-> E.y e. A ph)
Theorem	ceqsalt 2574*	Closed theorem version of ceqsalg 2576. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
|- (( F/xps /\ A.x(x = A -> (ph <-> ps)) /\ A e. V) -> (A.x(x = A -> ph) <-> ps))
Theorem	ceqsralt 2575*	Restricted quantifier version of ceqsalt 2574. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
|- (( F/xps /\ A.x(x = A -> (ph <-> ps)) /\ A e. B) -> (A.x e. B (x = A -> ph) <-> ps))
Theorem	ceqsalg 2576*	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/xps   &   |- (x = A -> (ph <-> ps))   =>   |- (A e. V -> (A.x(x = A -> ph) <-> ps))
Theorem	ceqsal 2577*	A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.)
|- F/xps   &   |- A e. _V   &   |- (x = A -> (ph <-> ps))   =>   |- (A.x(x = A -> ph) <-> ps)
Theorem	ceqsalv 2578*	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 2579*	Restricted quantifier version of ceqsalv 2578. (Contributed by NM, 21-Jun-2013.)
|- (x = A -> (ph <-> ps))   =>   |- (A e. B -> (A.x e. B (x = A -> ph) <-> ps))
Theorem	gencl 2580*	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 2581*	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 2582*	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 2583*	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 2584*	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.xE.y(ch /\ ph) <-> ps))
Theorem	cgsex4g 2585*	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.xE.yE.zE.w(ch /\ ph) <-> ps))
Theorem	ceqsex 2586*	Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) (Revised by Mario Carneiro, 10-Oct-2016.)
|- F/xps   &   |- A e. _V   &   |- (x = A -> (ph <-> ps))   =>   |- (E.x(x = A /\ ph) <-> ps)
Theorem	ceqsexv 2587*	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 2588*	Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.)
|- F/xps   &   |- F/ych   &   |- A e. _V   &   |- B e. _V   &   |- (x = A -> (ph <-> ps))   &   |- (y = B -> (ps <-> ch))   =>   |- (E.xE.y(x = A /\ y = B /\ ph) <-> ch)
Theorem	ceqsex2v 2589*	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.xE.y(x = A /\ y = B /\ ph) <-> ch)
Theorem	ceqsex3v 2590*	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.xE.yE.z((x = A /\ y = B /\ z = C) /\ ph) <-> th)
Theorem	ceqsex4v 2591*	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.xE.yE.zE.w((x = A /\ y = B) /\ (z = C /\ w = D) /\ ph) <-> ta)
Theorem	ceqsex6v 2592*	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.xE.yE.zE.wE.vE.u((x = A /\ y = B /\ z = C) /\ (w = D /\ v = E /\ u = F) /\ ph) <-> ze)
Theorem	ceqsex8v 2593*	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.xE.yE.zE.wE.vE.uE. tE. s(((x = A /\ y = B) /\ (z = C /\ w = D)) /\ ((v = E /\ u = F) /\ ( t = G /\ s = H)) /\ ph) <-> rh)
Theorem	gencbvex 2594*	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 2595*	Restatement of gencbvex 2594 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 2596*	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 2597*	Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf . (Contributed by Raph Levien, 10-Apr-2004.)
|- F/xps   &   |- (x = A -> (ph <-> ps))   =>   |- (y = A -> ([y / x]ph <-> ps))
Theorem	vtoclgft 2598	Closed theorem form of vtoclgf 2606. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 12-Oct-2016.)
|- ((( F/_xA /\ F/xps) /\ (A.x(x = A -> (ph <-> ps)) /\ A.xph) /\ A e. V) -> ps)
Theorem	vtocldf 2599	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/xph   &   |- (ph -> F/_xA)   &   |- (ph -> F/xch)   =>   |- (ph -> ch)
Theorem	vtocld 2600*	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)