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	cdeqab1 2901*	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 2902	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 2903	Conditional equality for set-to-class promotion. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- CondEq ( x = y -> x = y )
Theorem	cdeqeq 2904	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 2905	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 2906*	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 2907*	Variation of nfcdeq 2906 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 2908	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 4046. (Contributed by NM, 7-Aug-1994.)
|- { x | x e/ x } e/ _V
2.1.9  Proper substitution of classes for sets
Syntax	wsbc 2909	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 2910	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 2934 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 2911 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 2911, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 2910 in the form of sbc8g 2916. 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 2910 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 2911	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 2910 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 2912 instead of df-sbc 2910. (dfsbcq2 2912 is needed because unlike Quine we do not overload the df-sb 1736 syntax.) As a consequence of these theorems, we can derive sbc8g 2916, which is a weaker version of df-sbc 2910 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 2916, so we will allow direct use of df-sbc 2910. 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 2912	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 1736 and substitution for class variables df-sbc 2910. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 2911. (Contributed by NM, 31-Dec-2016.)
|- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
Theorem	sbsbc 2913	Show that df-sb 1736 and df-sbc 2910 are equivalent when the class term A in df-sbc 2910 is a setvar variable. This theorem lets us reuse theorems based on df-sb 1736 for proofs involving df-sbc 2910. (Contributed by NM, 31-Dec-2016.) (Proof modification is discouraged.)
|- ( [ y / x ] ph <-> [. y / x ]. ph )
Theorem	sbceq1d 2914	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 2915	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 2916	This is the closest we can get to df-sbc 2910 if we start from dfsbcq 2911 (see its comments) and dfsbcq2 2912. (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 2917	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 2918	Equality theorem for class substitution. Class version of sbequ12 1744. (Contributed by NM, 26-Sep-2003.)
|- ( x = A -> ( ph <-> [. A / x ]. ph ) )
Theorem	sbceq2a 2919	Equality theorem for class substitution. Class version of sbequ12r 1745. (Contributed by NM, 4-Jan-2017.)
|- ( A = x -> ( [. A / x ]. ph <-> ph ) )
Theorem	spsbc 2920	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 1748 and rspsbc 2991. (Contributed by NM, 16-Jan-2004.)
|- ( A e. V -> ( A. x ph -> [. A / x ]. ph ) )
Theorem	spsbcd 2921	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 1748 and rspsbc 2991. (Contributed by Mario Carneiro, 9-Feb-2017.)
|- ( ph -> A e. V )   &    |- ( ph -> A. x ps )   =>    |- ( ph -> [. A / x ]. ps )
Theorem	sbcth 2922	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 2923*	Deduction version of sbcth 2922. (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 2924	An identity theorem for substitution. See sbid 1747. (Contributed by Mario Carneiro, 18-Feb-2017.)
|- ( [. x / x ]. ph <-> ph )
Theorem	nfsbc1d 2925	Deduction version of nfsbc1 2926. (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 2926	Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.)
|- F/_ x A   =>    |- F/ x [. A / x ]. ph
Theorem	nfsbc1v 2927*	Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.)
|- F/ x [. A / x ]. ph
Theorem	nfsbcd 2928	Deduction version of nfsbc 2929. (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 2929	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 2930*	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 2931*	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 2932*	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 2933*	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 2934*	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 2935*	An equivalence for class substitution in the spirit of df-clab 2126. 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	cbvsbcw 2936*	Version of cbvsbc 2937 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.)
|- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( [. A / x ]. ph <-> [. A / y ]. ps )
Theorem	cbvsbc 2937	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 2938*	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 2939*	Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 2940.) (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 2940*	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 2941*	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 2942*	Conversion of implicit substitution to explicit class substitution. This version of sbcie 2943 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 2943*	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 2944*	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 2945*	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 2946*	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 ) )
Theorem	elrabsf 2947	Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 2838 has implicit substitution). The hypothesis specifies that x must not be a free variable in B. (Contributed by NM, 30-Sep-2003.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
|- F/_ x B   =>    |- ( A e. { x e. B | ph } <-> ( A e. B /\ [. A / x ]. ph ) )
Theorem	eqsbc3 2948*	Substitution applied to an atomic wff. Set theory version of eqsb3 2243. (Contributed by Andrew Salmon, 29-Jun-2011.)
|- ( A e. V -> ( [. A / x ]. x = B <-> A = B ) )
Theorem	sbcng 2949	Move negation in and out of class substitution. (Contributed by NM, 16-Jan-2004.)
|- ( A e. V -> ( [. A / x ]. -. ph <-> -. [. A / x ]. ph ) )
Theorem	sbcimg 2950	Distribution of class substitution over implication. (Contributed by NM, 16-Jan-2004.)
|- ( A e. V -> ( [. A / x ]. ( ph -> ps ) <-> ( [. A / x ]. ph -> [. A / x ]. ps ) ) )
Theorem	sbcan 2951	Distribution of class substitution over conjunction. (Contributed by NM, 31-Dec-2016.)
|- ( [. A / x ]. ( ph /\ ps ) <-> ( [. A / x ]. ph /\ [. A / x ]. ps ) )
Theorem	sbcang 2952	Distribution of class substitution over conjunction. (Contributed by NM, 21-May-2004.)
|- ( A e. V -> ( [. A / x ]. ( ph /\ ps ) <-> ( [. A / x ]. ph /\ [. A / x ]. ps ) ) )
Theorem	sbcor 2953	Distribution of class substitution over disjunction. (Contributed by NM, 31-Dec-2016.)
|- ( [. A / x ]. ( ph \/ ps ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps ) )
Theorem	sbcorg 2954	Distribution of class substitution over disjunction. (Contributed by NM, 21-May-2004.)
|- ( A e. V -> ( [. A / x ]. ( ph \/ ps ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps ) ) )
Theorem	sbcbig 2955	Distribution of class substitution over biconditional. (Contributed by Raph Levien, 10-Apr-2004.)
|- ( A e. V -> ( [. A / x ]. ( ph <-> ps ) <-> ( [. A / x ]. ph <-> [. A / x ]. ps ) ) )
Theorem	sbcn1 2956	Move negation in and out of class substitution. One direction of sbcng 2949 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)
|- ( [. A / x ]. -. ph -> -. [. A / x ]. ph )
Theorem	sbcim1 2957	Distribution of class substitution over implication. One direction of sbcimg 2950 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)
|- ( [. A / x ]. ( ph -> ps ) -> ( [. A / x ]. ph -> [. A / x ]. ps ) )
Theorem	sbcbi1 2958	Distribution of class substitution over biconditional. One direction of sbcbig 2955 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)
|- ( [. A / x ]. ( ph <-> ps ) -> ( [. A / x ]. ph <-> [. A / x ]. ps ) )
Theorem	sbcbi2 2959	Substituting into equivalent wff's gives equivalent results. (Contributed by Giovanni Mascellani, 9-Apr-2018.)
|- ( A. x ( ph <-> ps ) -> ( [. A / x ]. ph <-> [. A / x ]. ps ) )
Theorem	sbcal 2960*	Move universal quantifier in and out of class substitution. (Contributed by NM, 31-Dec-2016.)
|- ( [. A / y ]. A. x ph <-> A. x [. A / y ]. ph )
Theorem	sbcalg 2961*	Move universal quantifier in and out of class substitution. (Contributed by NM, 16-Jan-2004.)
|- ( A e. V -> ( [. A / y ]. A. x ph <-> A. x [. A / y ]. ph ) )
Theorem	sbcex2 2962*	Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.)
|- ( [. A / y ]. E. x ph <-> E. x [. A / y ]. ph )
Theorem	sbcexg 2963*	Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.)
|- ( A e. V -> ( [. A / y ]. E. x ph <-> E. x [. A / y ]. ph ) )
Theorem	sbceqal 2964*	A variation of extensionality for classes. (Contributed by Andrew Salmon, 28-Jun-2011.)
|- ( A e. V -> ( A. x ( x = A -> x = B ) -> A = B ) )
Theorem	sbeqalb 2965*	Theorem *14.121 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 28-Jun-2011.) (Proof shortened by Wolf Lammen, 9-May-2013.)
|- ( A e. V -> ( ( A. x ( ph <-> x = A ) /\ A. x ( ph <-> x = B ) ) -> A = B ) )
Theorem	sbcbid 2966	Formula-building deduction for class substitution. (Contributed by NM, 29-Dec-2014.)
|- F/ x ph   &    |- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( [. A / x ]. ps <-> [. A / x ]. ch ) )
Theorem	sbcbidv 2967*	Formula-building deduction for class substitution. (Contributed by NM, 29-Dec-2014.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( [. A / x ]. ps <-> [. A / x ]. ch ) )
Theorem	sbcbii 2968	Formula-building inference for class substitution. (Contributed by NM, 11-Nov-2005.)
|- ( ph <-> ps )   =>    |- ( [. A / x ]. ph <-> [. A / x ]. ps )
Theorem	eqsbc3r 2969*	eqsbc3 2948 with setvar variable on right side of equals sign. (Contributed by Alan Sare, 24-Oct-2011.) (Proof shortened by JJ, 7-Jul-2021.)
|- ( A e. V -> ( [. A / x ]. B = x <-> B = A ) )
Theorem	sbc3an 2970	Distribution of class substitution over triple conjunction. (Contributed by NM, 14-Dec-2006.) (Revised by NM, 17-Aug-2018.)
|- ( [. A / x ]. ( ph /\ ps /\ ch ) <-> ( [. A / x ]. ph /\ [. A / x ]. ps /\ [. A / x ]. ch ) )
Theorem	sbcel1v 2971*	Class substitution into a membership relation. (Contributed by NM, 17-Aug-2018.)
|- ( [. A / x ]. x e. B <-> A e. B )
Theorem	sbcel2gv 2972*	Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- ( B e. V -> ( [. B / x ]. A e. x <-> A e. B ) )
Theorem	sbcel21v 2973*	Class substitution into a membership relation. One direction of sbcel2gv 2972 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)
|- ( [. B / x ]. A e. x -> A e. B )
Theorem	sbcimdv 2974*	Substitution analogue of Theorem 19.20 of [Margaris] p. 90 (alim 1433). (Contributed by NM, 11-Nov-2005.) (Revised by NM, 17-Aug-2018.) (Proof shortened by JJ, 7-Jul-2021.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) )
Theorem	sbctt 2975	Substitution for a variable not free in a wff does not affect it. (Contributed by Mario Carneiro, 14-Oct-2016.)
|- ( ( A e. V /\ F/ x ph ) -> ( [. A / x ]. ph <-> ph ) )
Theorem	sbcgf 2976	Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- F/ x ph   =>    |- ( A e. V -> ( [. A / x ]. ph <-> ph ) )
Theorem	sbc19.21g 2977	Substitution for a variable not free in antecedent affects only the consequent. (Contributed by NM, 11-Oct-2004.)
|- F/ x ph   =>    |- ( A e. V -> ( [. A / x ]. ( ph -> ps ) <-> ( ph -> [. A / x ]. ps ) ) )
Theorem	sbcg 2978*	Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 2976. (Contributed by Alan Sare, 10-Nov-2012.)
|- ( A e. V -> ( [. A / x ]. ph <-> ph ) )
Theorem	sbc2iegf 2979*	Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Dec-2013.)
|- F/ x ps   &    |- F/ y ps   &    |- F/ x B e. W   &    |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )   =>    |- ( ( A e. V /\ B e. W ) -> ( [. A / x ]. [. B / y ]. ph <-> ps ) )
Theorem	sbc2ie 2980*	Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro, 19-Dec-2013.)
|- A e. _V   &    |- B e. _V   &    |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )   =>    |- ( [. A / x ]. [. B / y ]. ph <-> ps )
Theorem	sbc2iedv 2981*	Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Proof shortened by Mario Carneiro, 18-Oct-2016.)
|- A e. _V   &    |- B e. _V   &    |- ( ph -> ( ( x = A /\ y = B ) -> ( ps <-> ch ) ) )   =>    |- ( ph -> ( [. A / x ]. [. B / y ]. ps <-> ch ) )
Theorem	sbc3ie 2982*	Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Jun-2014.) (Revised by Mario Carneiro, 29-Dec-2014.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> ps ) )   =>    |- ( [. A / x ]. [. B / y ]. [. C / z ]. ph <-> ps )
Theorem	sbccomlem 2983*	Lemma for sbccom 2984. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 18-Oct-2016.)
|- ( [. A / x ]. [. B / y ]. ph <-> [. B / y ]. [. A / x ]. ph )
Theorem	sbccom 2984*	Commutative law for double class substitution. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Mario Carneiro, 18-Oct-2016.)
|- ( [. A / x ]. [. B / y ]. ph <-> [. B / y ]. [. A / x ]. ph )
Theorem	sbcralt 2985*	Interchange class substitution and restricted quantifier. (Contributed by NM, 1-Mar-2008.) (Revised by David Abernethy, 22-Feb-2010.)
|- ( ( A e. V /\ F/_ y A ) -> ( [. A / x ]. A. y e. B ph <-> A. y e. B [. A / x ]. ph ) )
Theorem	sbcrext 2986*	Interchange class substitution and restricted existential quantifier. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
|- ( F/_ y A -> ( [. A / x ]. E. y e. B ph <-> E. y e. B [. A / x ]. ph ) )
Theorem	sbcralg 2987*	Interchange class substitution and restricted quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- ( A e. V -> ( [. A / x ]. A. y e. B ph <-> A. y e. B [. A / x ]. ph ) )
Theorem	sbcrex 2988*	Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005.) (Revised by NM, 18-Aug-2018.)
|- ( [. A / x ]. E. y e. B ph <-> E. y e. B [. A / x ]. ph )
Theorem	sbcreug 2989*	Interchange class substitution and restricted unique existential quantifier. (Contributed by NM, 24-Feb-2013.)
|- ( A e. V -> ( [. A / x ]. E! y e. B ph <-> E! y e. B [. A / x ]. ph ) )
Theorem	sbcabel 2990*	Interchange class substitution and class abstraction. (Contributed by NM, 5-Nov-2005.)
|- F/_ x B   =>    |- ( A e. V -> ( [. A / x ]. { y | ph } e. B <-> { y | [. A / x ]. ph } e. B ) )
Theorem	rspsbc 2991*	Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. This provides an axiom for a predicate calculus for a restricted domain. This theorem generalizes the unrestricted stdpc4 1748 and spsbc 2920. See also rspsbca 2992 and rspcsbela . (Contributed by NM, 17-Nov-2006.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
|- ( A e. B -> ( A. x e. B ph -> [. A / x ]. ph ) )
Theorem	rspsbca 2992*	Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. (Contributed by NM, 14-Dec-2005.)
|- ( ( A e. B /\ A. x e. B ph ) -> [. A / x ]. ph )
Theorem	rspesbca 2993*	Existence form of rspsbca 2992. (Contributed by NM, 29-Feb-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
|- ( ( A e. B /\ [. A / x ]. ph ) -> E. x e. B ph )
Theorem	spesbc 2994	Existence form of spsbc 2920. (Contributed by Mario Carneiro, 18-Nov-2016.)
|- ( [. A / x ]. ph -> E. x ph )
Theorem	spesbcd 2995	form of spsbc 2920. (Contributed by Mario Carneiro, 9-Feb-2017.)
|- ( ph -> [. A / x ]. ps )   =>    |- ( ph -> E. x ps )
Theorem	sbcth2 2996*	A substitution into a theorem. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
|- ( x e. B -> ph )   =>    |- ( A e. B -> [. A / x ]. ph )
Theorem	ra5 2997	Restricted quantifier version of Axiom 5 of [Mendelson] p. 69. This is an axiom of a predicate calculus for a restricted domain. Compare the unrestricted stdpc5 1563. (Contributed by NM, 16-Jan-2004.)
|- F/ x ph   =>    |- ( A. x e. A ( ph -> ps ) -> ( ph -> A. x e. A ps ) )
Theorem	rmo2ilem 2998*	Condition implying restricted at-most-one quantifier. (Contributed by Jim Kingdon, 14-Jul-2018.)
|- F/ y ph   =>    |- ( E. y A. x e. A ( ph -> x = y ) -> E* x e. A ph )
Theorem	rmo2i 2999*	Condition implying restricted at-most-one quantifier. (Contributed by NM, 17-Jun-2017.)
|- F/ y ph   =>    |- ( E. y e. A A. x e. A ( ph -> x = y ) -> E* x e. A ph )
Theorem	rmo3 3000*	Restricted at-most-one quantifier using explicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.)
|- F/ y ph   =>    |- ( E* x e. A ph <-> A. x e. A A. y e. A ( ( ph /\ [ y / x ] ph ) -> x = y ) )