P. 30 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 2901-3000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ceqsexgv 2901*	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 2902*	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 2903*	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 2904*	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 2905*	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 2906*	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 2907*	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 2908*	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	clel5 2909*	Alternate definition of class membership: a class X is an element of another class A iff there is an element of A equal to X. (Contributed by AV, 13-Nov-2020.) (Revised by Steven Nguyen, 19-May-2023.)
|- ( X e. A <-> E. x e. A X = x )
Theorem	pm13.183 2910*	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 2911*	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 2912*	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 2913*	Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 2918.) (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 2914	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 2915*	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 2916*	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	elabd 2917*	Explicit demonstration the class { x | ps } is not empty by the example X. (Contributed by RP, 12-Aug-2020.)
|- ( ph -> X e. _V )   &    |- ( ph -> ch )   &    |- ( x = X -> ( ps <-> ch ) )   =>    |- ( ph -> E. x ps )
Theorem	elabg 2918*	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 2919*	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 2920*	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 2921*	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 2922	Membership in a class abstraction, with a weaker antecedent than elabgf 2914. (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 2923*	Membership in a class abstraction, with a weaker antecedent than elabg 2918. (Contributed by NM, 29-Aug-2006.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( ( ps -> A e. B ) -> ( A e. { x | ph } <-> ps ) )
Theorem	elab3 2924*	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 2925*	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 2926	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 2927*	Membership in a restricted class abstraction, using implicit substitution. (Closed theorem version of elrab3 2929.) (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 2928*	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 2929*	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	elrabd 2930*	Membership in a restricted class abstraction, using implicit substitution. Deduction version of elrab 2928. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
|- ( x = A -> ( ps <-> ch ) )   &    |- ( ph -> A e. B )   &    |- ( ph -> ch )   =>    |- ( ph -> A e. { x e. B | ps } )
Theorem	elrab2 2931*	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 ) )
Theorem	ralab 2932*	Universal quantification over a class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)
|- ( y = x -> ( ph <-> ps ) )   =>    |- ( A. x e. { y | ph } ch <-> A. x ( ps -> ch ) )
Theorem	ralrab 2933*	Universal quantification over a restricted class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)
|- ( y = x -> ( ph <-> ps ) )   =>    |- ( A. x e. { y e. A | ph } ch <-> A. x e. A ( ps -> ch ) )
Theorem	rexab 2934*	Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 23-Jan-2014.) (Revised by Mario Carneiro, 3-Sep-2015.)
|- ( y = x -> ( ph <-> ps ) )   =>    |- ( E. x e. { y | ph } ch <-> E. x ( ps /\ ch ) )
Theorem	rexrab 2935*	Existential quantification over a class abstraction. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Mario Carneiro, 3-Sep-2015.)
|- ( y = x -> ( ph <-> ps ) )   =>    |- ( E. x e. { y e. A | ph } ch <-> E. x e. A ( ps /\ ch ) )
Theorem	ralab2 2936*	Universal quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
|- ( x = y -> ( ps <-> ch ) )   =>    |- ( A. x e. { y | ph } ps <-> A. y ( ph -> ch ) )
Theorem	ralrab2 2937*	Universal quantification over a restricted class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
|- ( x = y -> ( ps <-> ch ) )   =>    |- ( A. x e. { y e. A | ph } ps <-> A. y e. A ( ph -> ch ) )
Theorem	rexab2 2938*	Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
|- ( x = y -> ( ps <-> ch ) )   =>    |- ( E. x e. { y | ph } ps <-> E. y ( ph /\ ch ) )
Theorem	rexrab2 2939*	Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
|- ( x = y -> ( ps <-> ch ) )   =>    |- ( E. x e. { y e. A | ph } ps <-> E. y e. A ( ph /\ ch ) )
Theorem	abidnf 2940*	Identity used to create closed-form versions of bound-variable hypothesis builders for class expressions. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Mario Carneiro, 12-Oct-2016.)
|- ( F/_ x A -> { z | A. x z e. A } = A )
Theorem	dedhb 2941*	A deduction theorem for converting the inference |- F/_ x A => |- ph into a closed theorem. Use nfa1 1563 and nfab 2352 to eliminate the hypothesis of the substitution instance ps of the inference. For converting the inference form into a deduction form, abidnf 2940 is useful. (Contributed by NM, 8-Dec-2006.)
|- ( A = { z | A. x z e. A } -> ( ph <-> ps ) )   &    |- ps   =>    |- ( F/_ x A -> ph )
Theorem	eqeu 2942*	A condition which implies existential uniqueness. (Contributed by Jeff Hankins, 8-Sep-2009.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( ( A e. B /\ ps /\ A. x ( ph -> x = A ) ) -> E! x ph )
Theorem	eueq 2943*	Equality has existential uniqueness. (Contributed by NM, 25-Nov-1994.)
|- ( A e. _V <-> E! x x = A )
Theorem	eueq1 2944*	Equality has existential uniqueness. (Contributed by NM, 5-Apr-1995.)
|- A e. _V   =>    |- E! x x = A
Theorem	eueq2dc 2945*	Equality has existential uniqueness (split into 2 cases). (Contributed by NM, 5-Apr-1995.)
|- A e. _V   &    |- B e. _V   =>    |- (DECID ph -> E! x ( ( ph /\ x = A ) \/ ( -. ph /\ x = B ) ) )
Theorem	eueq3dc 2946*	Equality has existential uniqueness (split into 3 cases). (Contributed by NM, 5-Apr-1995.) (Proof shortened by Mario Carneiro, 28-Sep-2015.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- -. ( ph /\ ps )   =>    |- (DECID ph -> (DECID ps -> E! x ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) ) )
Theorem	moeq 2947*	There is at most one set equal to a class. (Contributed by NM, 8-Mar-1995.)
|- E* x x = A
Theorem	moeq3dc 2948*	"At most one" property of equality (split into 3 cases). (Contributed by Jim Kingdon, 7-Jul-2018.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- -. ( ph /\ ps )   =>    |- (DECID ph -> (DECID ps -> E* x ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) ) )
Theorem	mosubt 2949*	"At most one" remains true after substitution. (Contributed by Jim Kingdon, 18-Jan-2019.)
|- ( A. y E* x ph -> E* x E. y ( y = A /\ ph ) )
Theorem	mosub 2950*	"At most one" remains true after substitution. (Contributed by NM, 9-Mar-1995.)
|- E* x ph   =>    |- E* x E. y ( y = A /\ ph )
Theorem	mo2icl 2951*	Theorem for inferring "at most one". (Contributed by NM, 17-Oct-1996.)
|- ( A. x ( ph -> x = A ) -> E* x ph )
Theorem	mob2 2952*	Consequence of "at most one". (Contributed by NM, 2-Jan-2015.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( ( A e. B /\ E* x ph /\ ph ) -> ( x = A <-> ps ) )
Theorem	moi2 2953*	Consequence of "at most one". (Contributed by NM, 29-Jun-2008.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( ( ( A e. B /\ E* x ph ) /\ ( ph /\ ps ) ) -> x = A )
Theorem	mob 2954*	Equality implied by "at most one". (Contributed by NM, 18-Feb-2006.)
|- ( x = A -> ( ph <-> ps ) )   &    |- ( x = B -> ( ph <-> ch ) )   =>    |- ( ( ( A e. C /\ B e. D ) /\ E* x ph /\ ps ) -> ( A = B <-> ch ) )
Theorem	moi 2955*	Equality implied by "at most one". (Contributed by NM, 18-Feb-2006.)
|- ( x = A -> ( ph <-> ps ) )   &    |- ( x = B -> ( ph <-> ch ) )   =>    |- ( ( ( A e. C /\ B e. D ) /\ E* x ph /\ ( ps /\ ch ) ) -> A = B )
Theorem	morex 2956*	Derive membership from uniqueness. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- B e. _V   &    |- ( x = B -> ( ph <-> ps ) )   =>    |- ( ( E. x e. A ph /\ E* x ph ) -> ( ps -> B e. A ) )
Theorem	euxfr2dc 2957*	Transfer existential uniqueness from a variable x to another variable y contained in expression A. (Contributed by NM, 14-Nov-2004.)
|- A e. _V   &    |- E* y x = A   =>    |- (DECID E. y E. x ( x = A /\ ph ) -> ( E! x E. y ( x = A /\ ph ) <-> E! y ph ) )
Theorem	euxfrdc 2958*	Transfer existential uniqueness from a variable x to another variable y contained in expression A. (Contributed by NM, 14-Nov-2004.)
|- A e. _V   &    |- E! y x = A   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- (DECID E. y E. x ( x = A /\ ps ) -> ( E! x ph <-> E! y ps ) )
Theorem	euind 2959*	Existential uniqueness via an indirect equality. (Contributed by NM, 11-Oct-2010.)
|- B e. _V   &    |- ( x = y -> ( ph <-> ps ) )   &    |- ( x = y -> A = B )   =>    |- ( ( A. x A. y ( ( ph /\ ps ) -> A = B ) /\ E. x ph ) -> E! z A. x ( ph -> z = A ) )
Theorem	reu2 2960*	A way to express restricted uniqueness. (Contributed by NM, 22-Nov-1994.)
|- ( E! x e. A ph <-> ( E. x e. A ph /\ A. x e. A A. y e. A ( ( ph /\ [ y / x ] ph ) -> x = y ) ) )
Theorem	reu6 2961*	A way to express restricted uniqueness. (Contributed by NM, 20-Oct-2006.)
|- ( E! x e. A ph <-> E. y e. A A. x e. A ( ph <-> x = y ) )
Theorem	reu3 2962*	A way to express restricted uniqueness. (Contributed by NM, 24-Oct-2006.)
|- ( E! x e. A ph <-> ( E. x e. A ph /\ E. y e. A A. x e. A ( ph -> x = y ) ) )
Theorem	reu6i 2963*	A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.)
|- ( ( B e. A /\ A. x e. A ( ph <-> x = B ) ) -> E! x e. A ph )
Theorem	eqreu 2964*	A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.)
|- ( x = B -> ( ph <-> ps ) )   =>    |- ( ( B e. A /\ ps /\ A. x e. A ( ph -> x = B ) ) -> E! x e. A ph )
Theorem	rmo4 2965*	Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by NM, 16-Jun-2017.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( E* x e. A ph <-> A. x e. A A. y e. A ( ( ph /\ ps ) -> x = y ) )
Theorem	reu4 2966*	Restricted uniqueness using implicit substitution. (Contributed by NM, 23-Nov-1994.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( E! x e. A ph <-> ( E. x e. A ph /\ A. x e. A A. y e. A ( ( ph /\ ps ) -> x = y ) ) )
Theorem	reu7 2967*	Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( E! x e. A ph <-> ( E. x e. A ph /\ E. x e. A A. y e. A ( ps -> x = y ) ) )
Theorem	reu8 2968*	Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( E! x e. A ph <-> E. x e. A ( ph /\ A. y e. A ( ps -> x = y ) ) )
Theorem	rmo3f 2969*	Restricted "at most one" using explicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)
|- F/_ x A   &    |- F/_ y A   &    |- F/ y ph   =>    |- ( E* x e. A ph <-> A. x e. A A. y e. A ( ( ph /\ [ y / x ] ph ) -> x = y ) )
Theorem	rmo4f 2970*	Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by Thierry Arnoux, 11-Oct-2016.) (Revised by Thierry Arnoux, 8-Mar-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)
|- F/_ x A   &    |- F/_ y A   &    |- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( E* x e. A ph <-> A. x e. A A. y e. A ( ( ph /\ ps ) -> x = y ) )
Theorem	reueq 2971*	Equality has existential uniqueness. (Contributed by Mario Carneiro, 1-Sep-2015.)
|- ( B e. A <-> E! x e. A x = B )
Theorem	rmoan 2972	Restricted "at most one" still holds when a conjunct is added. (Contributed by NM, 16-Jun-2017.)
|- ( E* x e. A ph -> E* x e. A ( ps /\ ph ) )
Theorem	rmoim 2973	Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.)
|- ( A. x e. A ( ph -> ps ) -> ( E* x e. A ps -> E* x e. A ph ) )
Theorem	rmoimia 2974	Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.)
|- ( x e. A -> ( ph -> ps ) )   =>    |- ( E* x e. A ps -> E* x e. A ph )
Theorem	rmoimi2 2975	Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.)
|- A. x ( ( x e. A /\ ph ) -> ( x e. B /\ ps ) )   =>    |- ( E* x e. B ps -> E* x e. A ph )
Theorem	2reuswapdc 2976*	A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by NM, 16-Jun-2017.)
|- (DECID E. x E. y ( x e. A /\ ( y e. B /\ ph ) ) -> ( A. x e. A E* y e. B ph -> ( E! x e. A E. y e. B ph -> E! y e. B E. x e. A ph ) ) )
Theorem	reuind 2977*	Existential uniqueness via an indirect equality. (Contributed by NM, 16-Oct-2010.)
|- ( x = y -> ( ph <-> ps ) )   &    |- ( x = y -> A = B )   =>    |- ( ( A. x A. y ( ( ( A e. C /\ ph ) /\ ( B e. C /\ ps ) ) -> A = B ) /\ E. x ( A e. C /\ ph ) ) -> E! z e. C A. x ( ( A e. C /\ ph ) -> z = A ) )
Theorem	2rmorex 2978*	Double restricted quantification with "at most one," analogous to 2moex 2139. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
|- ( E* x e. A E. y e. B ph -> A. y e. B E* x e. A ph )
Theorem	nelrdva 2979*	Deduce negative membership from an implication. (Contributed by Thierry Arnoux, 27-Nov-2017.)
|- ( ( ph /\ x e. A ) -> x =/= B )   =>    |- ( ph -> -. B e. A )
2.1.7  Conditional equality (experimental) This is a very useless definition, which "abbreviates" ( x = y -> ph ) as CondEq ( x = y -> ph ). What this display hides, though, is that the first expression, even though it has a shorter constant string, is actually much more complicated in its parse tree: it is parsed as (wi (wceq (cv vx) (cv vy)) wph), while the CondEq version is parsed as (wcdeq vx vy wph). It also allows us to give a name to the specific ternary operation ( x = y -> ph ). This is all used as part of a metatheorem: we want to say that |- ( x = y -> ( ph ( x ) <-> ph ( y ) ) ) and |- ( x = y -> A ( x ) = A ( y ) ) are provable, for any expressions ph ( x ) or A ( x ) in the language. The proof is by induction, so the base case is each of the primitives, which is why you will see a theorem for each of the set.mm primitive operations. The metatheorem comes with a disjoint variables condition: every variable in ph ( x ) is assumed disjoint from x except x itself. For such a proof by induction, we must consider each of the possible forms of ph ( x ). If it is a variable other than x, then we have CondEq ( x = y -> A = A ) or CondEq ( x = y -> ( ph <-> ph ) ), which is provable by cdeqth 2984 and reflexivity. Since we are only working with class and wff expressions, it can't be x itself in set.mm, but if it was we'd have to also prove CondEq ( x = y -> x = y ) (where set equality is being used on the right). Otherwise, it is a primitive operation applied to smaller expressions. In these cases, for each setvar variable parameter to the operation, we must consider if it is equal to x or not, which yields 2^n proof obligations. Luckily, all primitive operations in set.mm have either zero or one set variable, so we only need to prove one statement for the non-set constructors (like implication) and two for the constructors taking a set (the forall and the class builder). In each of the primitive proofs, we are allowed to assume that y is disjoint from ph ( x ) and vice versa, because this is maintained through the induction. This is how we satisfy the disjoint variable conditions of cdeqab1 2989 and cdeqab 2987.
Syntax	wcdeq 2980	Extend wff notation to include conditional equality. This is a technical device used in the proof that F/ is the not-free predicate, and that definitions are conservative as a result.
wff CondEq ( x = y -> ph )
Definition	df-cdeq 2981	Define conditional equality. All the notation to the left of the <-> is fake; the parentheses and arrows are all part of the notation, which could equally well be written CondEq x y ph. On the right side is the actual implication arrow. The reason for this definition is to "flatten" the structure on the right side (whose tree structure is something like (wi (wceq (cv vx) (cv vy)) wph) ) into just (wcdeq vx vy wph). (Contributed by Mario Carneiro, 11-Aug-2016.)
|- (CondEq ( x = y -> ph ) <-> ( x = y -> ph ) )
Theorem	cdeqi 2982	Deduce conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- ( x = y -> ph )   =>    |- CondEq ( x = y -> ph )
Theorem	cdeqri 2983	Property of conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- CondEq ( x = y -> ph )   =>    |- ( x = y -> ph )
Theorem	cdeqth 2984	Deduce conditional equality from a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- ph   =>    |- CondEq ( x = y -> ph )
Theorem	cdeqnot 2985	Distribute conditional equality over negation. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- CondEq ( x = y -> ( ph <-> ps ) )   =>    |- CondEq ( x = y -> ( -. ph <-> -. ps ) )
Theorem	cdeqal 2986*	Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- CondEq ( x = y -> ( ph <-> ps ) )   =>    |- CondEq ( x = y -> ( A. z ph <-> A. z ps ) )
Theorem	cdeqab 2987*	Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- CondEq ( x = y -> ( ph <-> ps ) )   =>    |- CondEq ( x = y -> { z | ph } = { z | ps } )
Theorem	cdeqal1 2988*	Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- CondEq ( x = y -> ( ph <-> ps ) )   =>    |- CondEq ( x = y -> ( A. x ph <-> A. y ps ) )
Theorem	cdeqab1 2989*	Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- CondEq ( x = y -> ( ph <-> ps ) )   =>    |- CondEq ( x = y -> { x | ph } = { y | ps } )
Theorem	cdeqim 2990	Distribute conditional equality over implication. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- CondEq ( x = y -> ( ph <-> ps ) )   &    |- CondEq ( x = y -> ( ch <-> th ) )   =>    |- CondEq ( x = y -> ( ( ph -> ch ) <-> ( ps -> th ) ) )
Theorem	cdeqcv 2991	Conditional equality for set-to-class promotion. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- CondEq ( x = y -> x = y )
Theorem	cdeqeq 2992	Distribute conditional equality over equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- CondEq ( x = y -> A = B )   &    |- CondEq ( x = y -> C = D )   =>    |- CondEq ( x = y -> ( A = C <-> B = D ) )
Theorem	cdeqel 2993	Distribute conditional equality over elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- CondEq ( x = y -> A = B )   &    |- CondEq ( x = y -> C = D )   =>    |- CondEq ( x = y -> ( A e. C <-> B e. D ) )
Theorem	nfcdeq 2994*	If we have a conditional equality proof, where ph is ph ( x ) and ps is ph ( y ), and ph ( x ) in fact does not have x free in it according to F/, then ph ( x ) <-> ph ( y ) unconditionally. This proves that F/ x ph is actually a not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/ x ph   &    |- CondEq ( x = y -> ( ph <-> ps ) )   =>    |- ( ph <-> ps )
Theorem	nfccdeq 2995*	Variation of nfcdeq 2994 for classes. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/_ x A   &    |- CondEq ( x = y -> A = B )   =>    |- A = B
2.1.8  Russell's Paradox
Theorem	ru 2996	Russell's Paradox. Proposition 4.14 of [TakeutiZaring] p. 14. In the late 1800s, Frege's Axiom of (unrestricted) Comprehension, expressed in our notation as A e. _V, asserted that any collection of sets A is a set i.e. belongs to the universe _V of all sets. In particular, by substituting { x | x e/ x } (the "Russell class") for A, it asserted { x | x e/ x } e. _V, meaning that the "collection of all sets which are not members of themselves" is a set. However, here we prove { x | x e/ x } e/ _V. This contradiction was discovered by Russell in 1901 (published in 1903), invalidating the Comprehension Axiom and leading to the collapse of Frege's system. In 1908, Zermelo rectified this fatal flaw by replacing Comprehension with a weaker Subset (or Separation) Axiom asserting that A is a set only when it is smaller than some other set B. The intuitionistic set theory IZF includes such a separation axiom, Axiom 6 of [Crosilla] p. "Axioms of CZF and IZF", which we include as ax-sep 4161. (Contributed by NM, 7-Aug-1994.)
|- { x | x e/ x } e/ _V
2.1.9  Proper substitution of classes for sets
Syntax	wsbc 2997	Extend wff notation to include the proper substitution of a class for a set. Read this notation as "the proper substitution of class A for setvar variable x in wff ph".
wff [. A / x ]. ph
Definition	df-sbc 2998	Define the proper substitution of a class for a set. When A is a proper class, our definition evaluates to false. This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3023 for our definition, which always evaluates to true for proper classes. Our definition also does not produce the same results as discussed in the proof of Theorem 6.6 of [Quine] p. 42 (although Theorem 6.6 itself does hold, as shown by dfsbcq 2999 below). Unfortunately, Quine's definition requires a recursive syntactical breakdown of ph, and it does not seem possible to express it with a single closed formula. If we did not want to commit to any specific proper class behavior, we could use this definition only to prove Theorem dfsbcq 2999, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 2998 in the form of sbc8g 3005. However, the behavior of Quine's definition at proper classes is similarly arbitrary, and for practical reasons (to avoid having to prove sethood of A in every use of this definition) we allow direct reference to df-sbc 2998 and assert that [. A / x ]. ph is always false when A is a proper class. The related definition df-csb defines proper substitution into a class variable (as opposed to a wff variable). (Contributed by NM, 14-Apr-1995.) (Revised by NM, 25-Dec-2016.)
|- ( [. A / x ]. ph <-> A e. { x | ph } )
Theorem	dfsbcq 2999	This theorem, which is similar to Theorem 6.7 of [Quine] p. 42 and holds under both our definition and Quine's, provides us with a weak definition of the proper substitution of a class for a set. Since our df-sbc 2998 does not result in the same behavior as Quine's for proper classes, if we wished to avoid conflict with Quine's definition we could start with this theorem and dfsbcq2 3000 instead of df-sbc 2998. (dfsbcq2 3000 is needed because unlike Quine we do not overload the df-sb 1785 syntax.) As a consequence of these theorems, we can derive sbc8g 3005, which is a weaker version of df-sbc 2998 that leaves substitution undefined when A is a proper class. However, it is often a nuisance to have to prove the sethood hypothesis of sbc8g 3005, so we will allow direct use of df-sbc 2998. Proper substiution with a proper class is rarely needed, and when it is, we can simply use the expansion of Quine's definition. (Contributed by NM, 14-Apr-1995.)
|- ( A = B -> ( [. A / x ]. ph <-> [. B / x ]. ph ) )
Theorem	dfsbcq2 3000	This theorem, which is similar to Theorem 6.7 of [Quine] p. 42 and holds under both our definition and Quine's, relates logic substitution df-sb 1785 and substitution for class variables df-sbc 2998. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 2999. (Contributed by NM, 31-Dec-2016.)
|- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )