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	3reeanv 2601*	Rearrange three existential quantifiers. (Contributed by Jeff Madsen, 11-Jun-2010.)
|- ( E. x e. A E. y e. B E. z e. C ( ph /\ ps /\ ch ) <-> ( E. x e. A ph /\ E. y e. B ps /\ E. z e. C ch ) )
Theorem	nfreu1 2602	x is not free in E! x e. A ph. (Contributed by NM, 19-Mar-1997.)
|- F/ x E! x e. A ph
Theorem	nfrmo1 2603	x is not free in E* x e. A ph. (Contributed by NM, 16-Jun-2017.)
|- F/ x E* x e. A ph
Theorem	nfreudxy 2604*	Not-free deduction for restricted uniqueness. This is a version where x and y are distinct. (Contributed by Jim Kingdon, 6-Jun-2018.)
|- F/ y ph   &    |- ( ph -> F/_ x A )   &    |- ( ph -> F/ x ps )   =>    |- ( ph -> F/ x E! y e. A ps )
Theorem	nfreuxy 2605*	Not-free for restricted uniqueness. This is a version where x and y are distinct. (Contributed by Jim Kingdon, 6-Jun-2018.)
|- F/_ x A   &    |- F/ x ph   =>    |- F/ x E! y e. A ph
Theorem	rabid 2606	An "identity" law of concretion for restricted abstraction. Special case of Definition 2.1 of [Quine] p. 16. (Contributed by NM, 9-Oct-2003.)
|- ( x e. { x e. A | ph } <-> ( x e. A /\ ph ) )
Theorem	rabid2 2607*	An "identity" law for restricted class abstraction. (Contributed by NM, 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)
|- ( A = { x e. A | ph } <-> A. x e. A ph )
Theorem	rabbi 2608	Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbidva 2674. (Contributed by NM, 25-Nov-2013.)
|- ( A. x e. A ( ps <-> ch ) <-> { x e. A | ps } = { x e. A | ch } )
Theorem	rabswap 2609	Swap with a membership relation in a restricted class abstraction. (Contributed by NM, 4-Jul-2005.)
|- { x e. A | x e. B } = { x e. B | x e. A }
Theorem	nfrab1 2610	The abstraction variable in a restricted class abstraction isn't free. (Contributed by NM, 19-Mar-1997.)
|- F/_ x { x e. A | ph }
Theorem	nfrabxy 2611*	A variable not free in a wff remains so in a restricted class abstraction. (Contributed by Jim Kingdon, 19-Jul-2018.)
|- F/ x ph   &    |- F/_ x A   =>    |- F/_ x { y e. A | ph }
Theorem	reubida 2612	Formula-building rule for restricted existential quantifier (deduction form). (Contributed by Mario Carneiro, 19-Nov-2016.)
|- F/ x ph   &    |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( E! x e. A ps <-> E! x e. A ch ) )
Theorem	reubidva 2613*	Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 13-Nov-2004.)
|- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( E! x e. A ps <-> E! x e. A ch ) )
Theorem	reubidv 2614*	Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 17-Oct-1996.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( E! x e. A ps <-> E! x e. A ch ) )
Theorem	reubiia 2615	Formula-building rule for restricted existential quantifier (inference form). (Contributed by NM, 14-Nov-2004.)
|- ( x e. A -> ( ph <-> ps ) )   =>    |- ( E! x e. A ph <-> E! x e. A ps )
Theorem	reubii 2616	Formula-building rule for restricted existential quantifier (inference form). (Contributed by NM, 22-Oct-1999.)
|- ( ph <-> ps )   =>    |- ( E! x e. A ph <-> E! x e. A ps )
Theorem	rmobida 2617	Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 16-Jun-2017.)
|- F/ x ph   &    |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( E* x e. A ps <-> E* x e. A ch ) )
Theorem	rmobidva 2618*	Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 16-Jun-2017.)
|- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( E* x e. A ps <-> E* x e. A ch ) )
Theorem	rmobidv 2619*	Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 16-Jun-2017.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( E* x e. A ps <-> E* x e. A ch ) )
Theorem	rmobiia 2620	Formula-building rule for restricted existential quantifier (inference form). (Contributed by NM, 16-Jun-2017.)
|- ( x e. A -> ( ph <-> ps ) )   =>    |- ( E* x e. A ph <-> E* x e. A ps )
Theorem	rmobii 2621	Formula-building rule for restricted existential quantifier (inference form). (Contributed by NM, 16-Jun-2017.)
|- ( ph <-> ps )   =>    |- ( E* x e. A ph <-> E* x e. A ps )
Theorem	raleqf 2622	Equality theorem for restricted universal quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.) (Revised by Andrew Salmon, 11-Jul-2011.)
|- F/_ x A   &    |- F/_ x B   =>    |- ( A = B -> ( A. x e. A ph <-> A. x e. B ph ) )
Theorem	rexeqf 2623	Equality theorem for restricted existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 9-Oct-2003.) (Revised by Andrew Salmon, 11-Jul-2011.)
|- F/_ x A   &    |- F/_ x B   =>    |- ( A = B -> ( E. x e. A ph <-> E. x e. B ph ) )
Theorem	reueq1f 2624	Equality theorem for restricted unique existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 5-Apr-2004.) (Revised by Andrew Salmon, 11-Jul-2011.)
|- F/_ x A   &    |- F/_ x B   =>    |- ( A = B -> ( E! x e. A ph <-> E! x e. B ph ) )
Theorem	rmoeq1f 2625	Equality theorem for restricted at-most-one quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
|- F/_ x A   &    |- F/_ x B   =>    |- ( A = B -> ( E* x e. A ph <-> E* x e. B ph ) )
Theorem	raleq 2626*	Equality theorem for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.)
|- ( A = B -> ( A. x e. A ph <-> A. x e. B ph ) )
Theorem	rexeq 2627*	Equality theorem for restricted existential quantifier. (Contributed by NM, 29-Oct-1995.)
|- ( A = B -> ( E. x e. A ph <-> E. x e. B ph ) )
Theorem	reueq1 2628*	Equality theorem for restricted unique existential quantifier. (Contributed by NM, 5-Apr-2004.)
|- ( A = B -> ( E! x e. A ph <-> E! x e. B ph ) )
Theorem	rmoeq1 2629*	Equality theorem for restricted at-most-one quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
|- ( A = B -> ( E* x e. A ph <-> E* x e. B ph ) )
Theorem	raleqi 2630*	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 2631*	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 2632*	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 2633*	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 2634*	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 2635*	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 2636*	Equality deduction for restricted unique existential 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 2637*	Equality deduction for restricted at-most-one 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 2638*	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 2639*	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 2640*	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 2641*	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 2642	Unrestricted "at most one" implies restricted "at most one". (Contributed by NM, 16-Jun-2017.)
|- ( E* x ph -> E* x e. A ph )
Theorem	reu5 2643	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 2644	Restricted unique existence implies restricted existence. (Contributed by NM, 19-Aug-1999.)
|- ( E! x e. A ph -> E. x e. A ph )
Theorem	reurmo 2645	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 2646	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 2647	Nonexistence implies restricted "at most one". (Contributed by NM, 17-Jun-2017.)
|- ( -. E. x e. A ph -> E* x e. A ph )
Theorem	cbvralf 2648	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 7-Mar-2004.) (Revised by Mario Carneiro, 9-Oct-2016.)
|- F/_ x A   &    |- F/_ y A   &    |- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( A. x e. A ph <-> A. y e. A ps )
Theorem	cbvrexf 2649	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/_ x A   &    |- F/_ y A   &    |- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( E. x e. A ph <-> E. y e. A ps )
Theorem	cbvral 2650*	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.)
|- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( A. x e. A ph <-> A. y e. A ps )
Theorem	cbvrex 2651*	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
|- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( E. x e. A ph <-> E. y e. A ps )
Theorem	cbvreu 2652*	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 2653*	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 2654*	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 2655*	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 2656*	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 2657*	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 2658*	Version of cbvralv 2654 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( A. x e. A ph <-> A. y e. A ps )
Theorem	cbvrexvw 2659*	Version of cbvrexv 2655 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( E. x e. A ph <-> E. y e. A ps )
Theorem	cbvreuvw 2660*	Version of cbvreuv 2656 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( E! x e. A ph <-> E! y e. A ps )
Theorem	cbvraldva2 2661*	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 2662*	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 2663*	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 2664*	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 2665*	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 2666*	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 2667*	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 2668*	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 2669*	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 2670*	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 2671	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 2672	Equivalent wff's correspond to equal restricted class abstractions. Inference form of rabbidv 2675. (Contributed by Peter Mazsa, 1-Nov-2019.)
|- ( ph <-> ps )   =>    |- { x e. A | ph } = { x e. A | ps }
Theorem	rabbidva2 2673*	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 2674*	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 2675*	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 2676	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 2677	Equality theorem for restricted class abstractions. Inference form of rabeqf 2676. (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 2678*	Equality theorem for restricted class abstractions. (Contributed by NM, 15-Oct-2003.)
|- ( A = B -> { x e. A | ph } = { x e. B | ph } )
Theorem	rabeqi 2679*	Equality theorem for restricted class abstractions. Inference form of rabeq 2678. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
|- A = B   =>    |- { x e. A | ph } = { x e. B | ph }
Theorem	rabeqdv 2680*	Equality of restricted class abstractions. Deduction form of rabeq 2678. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- ( ph -> A = B )   =>    |- ( ph -> { x e. A | ps } = { x e. B | ps } )
Theorem	rabeqbidv 2681*	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 2682*	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 2683	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 2684	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 2685*	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 2686	Extend class notation to include the universal class symbol.
class _V
Theorem	vjust 2687	Soundness justification theorem for df-v 2688. (Contributed by Rodolfo Medina, 27-Apr-2010.)
|- { x | x = x } = { y | y = y }
Definition	df-v 2688	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 2689	All setvar variables are sets (see isset 2692). Theorem 6.8 of [Quine] p. 43. (Contributed by NM, 5-Aug-1993.)
|- x e. _V
Theorem	elv 2690	Technical lemma used to shorten proofs. If a proposition is implied by x e. _V (which is true, see vex 2689), then it is true. (Contributed by Peter Mazsa, 13-Oct-2018.)
|- ( x e. _V -> ph )   =>    |- ph
Theorem	elvd 2691	Technical lemma used to shorten proofs. If a proposition is implied by x e. _V (which is true, see vex 2689) 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 2692*	Two ways to say " A is a set": A class A is a member of the universal class _V (see df-v 2688) 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 4359. Note the when A is not a set, it is called a proper class. In some theorems, such as uniexg 4361, 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 2135 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 2693	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 2694*	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 2695*	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 2696	A class equal to a variable is a set. Note the absence of disjoint variable condition, contrary to isset 2692 and issetri 2695. (Contributed by BJ, 27-Apr-2019.)
|- ( x = A -> A e. _V )
Theorem	elex 2697	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 2698	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 2699	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 2700*	An element of a class exists. (Contributed by NM, 1-May-1995.)
|- ( A e. V -> E. x x = A )