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	elab4g 2901*	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 2902	Membership in a class abstraction, with a weaker antecedent than elabgf 2894. (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 2903*	Membership in a class abstraction, with a weaker antecedent than elabg 2898. (Contributed by NM, 29-Aug-2006.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( ( ps -> A e. B ) -> ( A e. { x | ph } <-> ps ) )
Theorem	elab3 2904*	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 2905*	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 2906	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 2907*	Membership in a restricted class abstraction, using implicit substitution. (Closed theorem version of elrab3 2909.) (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 2908*	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 2909*	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 2910*	Membership in a restricted class abstraction, using implicit substitution. Deduction version of elrab 2908. (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 2911*	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 2912*	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 2913*	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 2914*	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 2915*	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 2916*	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 2917*	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 2918*	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 2919*	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 2920*	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 2921*	A deduction theorem for converting the inference |- F/_ x A => |- ph into a closed theorem. Use nfa1 1552 and nfab 2337 to eliminate the hypothesis of the substitution instance ps of the inference. For converting the inference form into a deduction form, abidnf 2920 is useful. (Contributed by NM, 8-Dec-2006.)
|- ( A = { z | A. x z e. A } -> ( ph <-> ps ) )   &    |- ps   =>    |- ( F/_ x A -> ph )
Theorem	eqeu 2922*	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 2923*	Equality has existential uniqueness. (Contributed by NM, 25-Nov-1994.)
|- ( A e. _V <-> E! x x = A )
Theorem	eueq1 2924*	Equality has existential uniqueness. (Contributed by NM, 5-Apr-1995.)
|- A e. _V   =>    |- E! x x = A
Theorem	eueq2dc 2925*	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 2926*	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 2927*	There is at most one set equal to a class. (Contributed by NM, 8-Mar-1995.)
|- E* x x = A
Theorem	moeq3dc 2928*	"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 2929*	"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 2930*	"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 2931*	Theorem for inferring "at most one". (Contributed by NM, 17-Oct-1996.)
|- ( A. x ( ph -> x = A ) -> E* x ph )
Theorem	mob2 2932*	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 2933*	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 2934*	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 2935*	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 2936*	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 2937*	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 2938*	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 2939*	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 2940*	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 2941*	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 2942*	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 2943*	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 2944*	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 2945*	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 2946*	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 2947*	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 2948*	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 2949*	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 2950*	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 2951*	Equality has existential uniqueness. (Contributed by Mario Carneiro, 1-Sep-2015.)
|- ( B e. A <-> E! x e. A x = B )
Theorem	rmoan 2952	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 2953	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 2954	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 2955	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 2956*	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 2957*	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 2958*	Double restricted quantification with "at most one," analogous to 2moex 2124. (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 2959*	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 2964 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 2969 and cdeqab 2967.
Syntax	wcdeq 2960	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 2961	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 2962	Deduce conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- ( x = y -> ph )   =>    |- CondEq ( x = y -> ph )
Theorem	cdeqri 2963	Property of conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- CondEq ( x = y -> ph )   =>    |- ( x = y -> ph )
Theorem	cdeqth 2964	Deduce conditional equality from a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- ph   =>    |- CondEq ( x = y -> ph )
Theorem	cdeqnot 2965	Distribute conditional equality over negation. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- CondEq ( x = y -> ( ph <-> ps ) )   =>    |- CondEq ( x = y -> ( -. ph <-> -. ps ) )
Theorem	cdeqal 2966*	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 2967*	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 2968*	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 2969*	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 2970	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 2971	Conditional equality for set-to-class promotion. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- CondEq ( x = y -> x = y )
Theorem	cdeqeq 2972	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 2973	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 2974*	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 2975*	Variation of nfcdeq 2974 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 2976	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 4136. (Contributed by NM, 7-Aug-1994.)
|- { x | x e/ x } e/ _V
2.1.9  Proper substitution of classes for sets
Syntax	wsbc 2977	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 2978	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 3003 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 2979 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 2979, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 2978 in the form of sbc8g 2985. 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 2978 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 2979	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 2978 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 2980 instead of df-sbc 2978. (dfsbcq2 2980 is needed because unlike Quine we do not overload the df-sb 1774 syntax.) As a consequence of these theorems, we can derive sbc8g 2985, which is a weaker version of df-sbc 2978 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 2985, so we will allow direct use of df-sbc 2978. 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 2980	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 1774 and substitution for class variables df-sbc 2978. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 2979. (Contributed by NM, 31-Dec-2016.)
|- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
Theorem	sbsbc 2981	Show that df-sb 1774 and df-sbc 2978 are equivalent when the class term A in df-sbc 2978 is a setvar variable. This theorem lets us reuse theorems based on df-sb 1774 for proofs involving df-sbc 2978. (Contributed by NM, 31-Dec-2016.) (Proof modification is discouraged.)
|- ( [ y / x ] ph <-> [. y / x ]. ph )
Theorem	sbceq1d 2982	Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.)
|- ( ph -> A = B )   =>    |- ( ph -> ( [. A / x ]. ps <-> [. B / x ]. ps ) )
Theorem	sbceq1dd 2983	Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.)
|- ( ph -> A = B )   &    |- ( ph -> [. A / x ]. ps )   =>    |- ( ph -> [. B / x ]. ps )
Theorem	sbceqbid 2984*	Equality theorem for class substitution. (Contributed by Thierry Arnoux, 4-Sep-2018.)
|- ( ph -> A = B )   &    |- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( [. A / x ]. ps <-> [. B / x ]. ch ) )
Theorem	sbc8g 2985	This is the closest we can get to df-sbc 2978 if we start from dfsbcq 2979 (see its comments) and dfsbcq2 2980. (Contributed by NM, 18-Nov-2008.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.)
|- ( A e. V -> ( [. A / x ]. ph <-> A e. { x | ph } ) )
Theorem	sbcex 2986	By our definition of proper substitution, it can only be true if the substituted expression is a set. (Contributed by Mario Carneiro, 13-Oct-2016.)
|- ( [. A / x ]. ph -> A e. _V )
Theorem	sbceq1a 2987	Equality theorem for class substitution. Class version of sbequ12 1782. (Contributed by NM, 26-Sep-2003.)
|- ( x = A -> ( ph <-> [. A / x ]. ph ) )
Theorem	sbceq2a 2988	Equality theorem for class substitution. Class version of sbequ12r 1783. (Contributed by NM, 4-Jan-2017.)
|- ( A = x -> ( [. A / x ]. ph <-> ph ) )
Theorem	spsbc 2989	Specialization: if a formula is true for all sets, it is true for any class which is a set. Similar to Theorem 6.11 of [Quine] p. 44. See also stdpc4 1786 and rspsbc 3060. (Contributed by NM, 16-Jan-2004.)
|- ( A e. V -> ( A. x ph -> [. A / x ]. ph ) )
Theorem	spsbcd 2990	Specialization: if a formula is true for all sets, it is true for any class which is a set. Similar to Theorem 6.11 of [Quine] p. 44. See also stdpc4 1786 and rspsbc 3060. (Contributed by Mario Carneiro, 9-Feb-2017.)
|- ( ph -> A e. V )   &    |- ( ph -> A. x ps )   =>    |- ( ph -> [. A / x ]. ps )
Theorem	sbcth 2991	A substitution into a theorem remains true (when A is a set). (Contributed by NM, 5-Nov-2005.)
|- ph   =>    |- ( A e. V -> [. A / x ]. ph )
Theorem	sbcthdv 2992*	Deduction version of sbcth 2991. (Contributed by NM, 30-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
|- ( ph -> ps )   =>    |- ( ( ph /\ A e. V ) -> [. A / x ]. ps )
Theorem	sbcid 2993	An identity theorem for substitution. See sbid 1785. (Contributed by Mario Carneiro, 18-Feb-2017.)
|- ( [. x / x ]. ph <-> ph )
Theorem	nfsbc1d 2994	Deduction version of nfsbc1 2995. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 12-Oct-2016.)
|- ( ph -> F/_ x A )   =>    |- ( ph -> F/ x [. A / x ]. ps )
Theorem	nfsbc1 2995	Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.)
|- F/_ x A   =>    |- F/ x [. A / x ]. ph
Theorem	nfsbc1v 2996*	Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.)
|- F/ x [. A / x ]. ph
Theorem	nfsbcd 2997	Deduction version of nfsbc 2998. (Contributed by NM, 23-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.)
|- F/ y ph   &    |- ( ph -> F/_ x A )   &    |- ( ph -> F/ x ps )   =>    |- ( ph -> F/ x [. A / y ]. ps )
Theorem	nfsbc 2998	Bound-variable hypothesis builder for class substitution. (Contributed by NM, 7-Sep-2014.) (Revised by Mario Carneiro, 12-Oct-2016.)
|- F/_ x A   &    |- F/ x ph   =>    |- F/ x [. A / y ]. ph
Theorem	sbcco 2999*	A composition law for class substitution. (Contributed by NM, 26-Sep-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)
|- ( [. A / y ]. [. y / x ]. ph <-> [. A / x ]. ph )
Theorem	sbcco2 3000*	A composition law for class substitution. Importantly, x may occur free in the class expression substituted for A. (Contributed by NM, 5-Sep-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
|- ( x = y -> A = B )   =>    |- ( [. x / y ]. [. B / x ]. ph <-> [. A / x ]. ph )