P. 28 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 2701-2800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ralab 2701*	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 2702*	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 2703*	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 2704*	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 2705*	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 2706*	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 2707*	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 2708*	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 2709*	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 2710*	A deduction theorem for converting the inference |- F/_ x A => |- ph into a closed theorem. Use nfa1 1434 and nfab 2182 to eliminate the hypothesis of the substitution instance ps of the inference. For converting the inference form into a deduction form, abidnf 2709 is useful. (Contributed by NM, 8-Dec-2006.)
|- ( A = { z | A. x z e. A } -> ( ph <-> ps ) )   &    |- ps   =>    |- ( F/_ x A -> ph )
Theorem	eqeu 2711*	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 2712*	Equality has existential uniqueness. (Contributed by NM, 25-Nov-1994.)
|- ( A e. _V <-> E! x x = A )
Theorem	eueq1 2713*	Equality has existential uniqueness. (Contributed by NM, 5-Apr-1995.)
|- A e. _V   =>    |- E! x x = A
Theorem	eueq2dc 2714*	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 2715*	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 2716*	There is at most one set equal to a class. (Contributed by NM, 8-Mar-1995.)
|- E* x x = A
Theorem	moeq3dc 2717*	"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 2718*	"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 2719*	"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 2720*	Theorem for inferring "at most one." (Contributed by NM, 17-Oct-1996.)
|- ( A. x ( ph -> x = A ) -> E* x ph )
Theorem	mob2 2721*	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 2722*	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 2723*	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 2724*	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 2725*	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 2726*	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 2727*	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 2728*	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 2729*	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 2730*	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 2731*	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 2732*	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 2733*	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 2734*	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 2735*	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 2736*	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 2737*	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	reueq 2738*	Equality has existential uniqueness. (Contributed by Mario Carneiro, 1-Sep-2015.)
|- ( B e. A <-> E! x e. A x = B )
Theorem	rmoan 2739	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 2740	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 2741	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 2742	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 2743*	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 2744*	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 2745*	Double restricted quantification with "at most one," analogous to 2moex 1986. (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 2746*	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 3-ary 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 assumption: 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 2751 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 DV assumptions of cdeqab1 2756 and cdeqab 2754.
Syntax	wcdeq 2747	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 2748	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 2749	Deduce conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- ( x = y -> ph )   =>    |- CondEq ( x = y -> ph )
Theorem	cdeqri 2750	Property of conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- CondEq ( x = y -> ph )   =>    |- ( x = y -> ph )
Theorem	cdeqth 2751	Deduce conditional equality from a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- ph   =>    |- CondEq ( x = y -> ph )
Theorem	cdeqnot 2752	Distribute conditional equality over negation. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- CondEq ( x = y -> ( ph <-> ps ) )   =>    |- CondEq ( x = y -> ( -. ph <-> -. ps ) )
Theorem	cdeqal 2753*	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 2754*	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 2755*	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 2756*	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 2757	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 2758	Conditional equality for set-to-class promotion. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- CondEq ( x = y -> x = y )
Theorem	cdeqeq 2759	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 2760	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 2761*	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 2762*	Variation of nfcdeq 2761 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 2763	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 3875. (Contributed by NM, 7-Aug-1994.)
|- { x | x e/ x } e/ _V
2.1.9  Proper substitution of classes for sets
Syntax	wsbc 2764	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 2765	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 2789 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 2766 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 2766, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 2765 in the form of sbc8g 2771. 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 2765 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 2766	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 2765 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 2767 instead of df-sbc 2765. (dfsbcq2 2767 is needed because unlike Quine we do not overload the df-sb 1646 syntax.) As a consequence of these theorems, we can derive sbc8g 2771, which is a weaker version of df-sbc 2765 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 2771, so we will allow direct use of df-sbc 2765. 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 2767	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 1646 and substitution for class variables df-sbc 2765. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 2766. (Contributed by NM, 31-Dec-2016.)
|- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
Theorem	sbsbc 2768	Show that df-sb 1646 and df-sbc 2765 are equivalent when the class term A in df-sbc 2765 is a setvar variable. This theorem lets us reuse theorems based on df-sb 1646 for proofs involving df-sbc 2765. (Contributed by NM, 31-Dec-2016.) (Proof modification is discouraged.)
|- ( [ y / x ] ph <-> [. y / x ]. ph )
Theorem	sbceq1d 2769	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 2770	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	sbc8g 2771	This is the closest we can get to df-sbc 2765 if we start from dfsbcq 2766 (see its comments) and dfsbcq2 2767. (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 2772	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 2773	Equality theorem for class substitution. Class version of sbequ12 1654. (Contributed by NM, 26-Sep-2003.)
|- ( x = A -> ( ph <-> [. A / x ]. ph ) )
Theorem	sbceq2a 2774	Equality theorem for class substitution. Class version of sbequ12r 1655. (Contributed by NM, 4-Jan-2017.)
|- ( A = x -> ( [. A / x ]. ph <-> ph ) )
Theorem	spsbc 2775	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 1658 and rspsbc 2840. (Contributed by NM, 16-Jan-2004.)
|- ( A e. V -> ( A. x ph -> [. A / x ]. ph ) )
Theorem	spsbcd 2776	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 1658 and rspsbc 2840. (Contributed by Mario Carneiro, 9-Feb-2017.)
|- ( ph -> A e. V )   &    |- ( ph -> A. x ps )   =>    |- ( ph -> [. A / x ]. ps )
Theorem	sbcth 2777	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 2778*	Deduction version of sbcth 2777. (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 2779	An identity theorem for substitution. See sbid 1657. (Contributed by Mario Carneiro, 18-Feb-2017.)
|- ( [. x / x ]. ph <-> ph )
Theorem	nfsbc1d 2780	Deduction version of nfsbc1 2781. (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 2781	Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.)
|- F/_ x A   =>    |- F/ x [. A / x ]. ph
Theorem	nfsbc1v 2782*	Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.)
|- F/ x [. A / x ]. ph
Theorem	nfsbcd 2783	Deduction version of nfsbc 2784. (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 2784	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 2785*	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 2786*	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 )
Theorem	sbc5 2787*	An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.)
|- ( [. A / x ]. ph <-> E. x ( x = A /\ ph ) )
Theorem	sbc6g 2788*	An equivalence for class substitution. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
|- ( A e. V -> ( [. A / x ]. ph <-> A. x ( x = A -> ph ) ) )
Theorem	sbc6 2789*	An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Proof shortened by Eric Schmidt, 17-Jan-2007.)
|- A e. _V   =>    |- ( [. A / x ]. ph <-> A. x ( x = A -> ph ) )
Theorem	sbc7 2790*	An equivalence for class substitution in the spirit of df-clab 2027. Note that x and A don't have to be distinct. (Contributed by NM, 18-Nov-2008.) (Revised by Mario Carneiro, 13-Oct-2016.)
|- ( [. A / x ]. ph <-> E. y ( y = A /\ [. y / x ]. ph ) )
Theorem	cbvsbc 2791	Change bound variables in a wff substitution. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
|- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( [. A / x ]. ph <-> [. A / y ]. ps )
Theorem	cbvsbcv 2792*	Change the bound variable of a class substitution using implicit substitution. (Contributed by NM, 30-Sep-2008.) (Revised by Mario Carneiro, 13-Oct-2016.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( [. A / x ]. ph <-> [. A / y ]. ps )
Theorem	sbciegft 2793*	Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 2794.) (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
|- ( ( A e. V /\ F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) ) -> ( [. A / x ]. ph <-> ps ) )
Theorem	sbciegf 2794*	Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
|- F/ x ps   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. V -> ( [. A / x ]. ph <-> ps ) )
Theorem	sbcieg 2795*	Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 10-Nov-2005.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. V -> ( [. A / x ]. ph <-> ps ) )
Theorem	sbcie2g 2796*	Conversion of implicit substitution to explicit class substitution. This version of sbcie 2797 avoids a disjointness condition on x and A by substituting twice. (Contributed by Mario Carneiro, 15-Oct-2016.)
|- ( x = y -> ( ph <-> ps ) )   &    |- ( y = A -> ( ps <-> ch ) )   =>    |- ( A e. V -> ( [. A / x ]. ph <-> ch ) )
Theorem	sbcie 2797*	Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 4-Sep-2004.)
|- A e. _V   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( [. A / x ]. ph <-> ps )
Theorem	sbciedf 2798*	Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 29-Dec-2014.)
|- ( ph -> A e. V )   &    |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )   &    |- F/ x ph   &    |- ( ph -> F/ x ch )   =>    |- ( ph -> ( [. A / x ]. ps <-> ch ) )
Theorem	sbcied 2799*	Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.)
|- ( ph -> A e. V )   &    |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( [. A / x ]. ps <-> ch ) )
Theorem	sbcied2 2800*	Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.)
|- ( ph -> A e. V )   &    |- ( ph -> A = B )   &    |- ( ( ph /\ x = B ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( [. A / x ]. ps <-> ch ) )