P. 32 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 3101-3200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	sbc2iedv 3101*	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 3102*	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 3103*	Lemma for sbccom 3104. (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 3104*	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 3105*	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 3106*	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 3107*	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 3108*	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 3109*	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	reu8nf 3110*	Restricted uniqueness using implicit substitution. This version of reu8 2999 uses a nonfreeness hypothesis for x and ps instead of distinct variable conditions. (Contributed by AV, 21-Jan-2022.)
|- F/ x ps   &    |- F/ x ch   &    |- ( x = w -> ( ph <-> ch ) )   &    |- ( w = y -> ( ch <-> ps ) )   =>    |- ( E! x e. A ph <-> E. x e. A ( ph /\ A. y e. A ( ps -> x = y ) ) )
Theorem	sbcabel 3111*	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 3112*	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 1821 and spsbc 3040. See also rspsbca 3113 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 3113*	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 3114*	Existence form of rspsbca 3113. (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 3115	Existence form of spsbc 3040. (Contributed by Mario Carneiro, 18-Nov-2016.)
|- ( [. A / x ]. ph -> E. x ph )
Theorem	spesbcd 3116	form of spsbc 3040. (Contributed by Mario Carneiro, 9-Feb-2017.)
|- ( ph -> [. A / x ]. ps )   =>    |- ( ph -> E. x ps )
Theorem	sbcth2 3117*	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 3118	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 1630. (Contributed by NM, 16-Jan-2004.)
|- F/ x ph   =>    |- ( A. x e. A ( ph -> ps ) -> ( ph -> A. x e. A ps ) )
Theorem	rmo2ilem 3119*	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 3120*	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 3121*	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 ) )
Theorem	rmob 3122*	Consequence of "at most one", using implicit substitution. (Contributed by NM, 2-Jan-2015.) (Revised by NM, 16-Jun-2017.)
|- ( x = B -> ( ph <-> ps ) )   &    |- ( x = C -> ( ph <-> ch ) )   =>    |- ( ( E* x e. A ph /\ ( B e. A /\ ps ) ) -> ( B = C <-> ( C e. A /\ ch ) ) )
Theorem	rmoi 3123*	Consequence of "at most one", using implicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.)
|- ( x = B -> ( ph <-> ps ) )   &    |- ( x = C -> ( ph <-> ch ) )   =>    |- ( ( E* x e. A ph /\ ( B e. A /\ ps ) /\ ( C e. A /\ ch ) ) -> B = C )
2.1.10  Proper substitution of classes for sets into classes
Syntax	csb 3124	Extend class notation to include the proper substitution of a class for a set into another class.
class [_ A / x ]_ B
Definition	df-csb 3125*	Define the proper substitution of a class for a set into another class. The underlined brackets distinguish it from the substitution into a wff, wsbc 3028, to prevent ambiguity. Theorem sbcel1g 3143 shows an example of how ambiguity could arise if we didn't use distinguished brackets. Theorem sbccsbg 3153 recreates substitution into a wff from this definition. (Contributed by NM, 10-Nov-2005.)
|- [_ A / x ]_ B = { y | [. A / x ]. y e. B }
Theorem	csb2 3126*	Alternate expression for the proper substitution into a class, without referencing substitution into a wff. Note that x can be free in B but cannot occur in A. (Contributed by NM, 2-Dec-2013.)
|- [_ A / x ]_ B = { y | E. x ( x = A /\ y e. B ) }
Theorem	csbeq1 3127	Analog of dfsbcq 3030 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
|- ( A = B -> [_ A / x ]_ C = [_ B / x ]_ C )
Theorem	cbvcsbw 3128*	Version of cbvcsb 3129 with a disjoint variable condition. (Contributed by GG, 10-Jan-2024.)
|- F/_ y C   &    |- F/_ x D   &    |- ( x = y -> C = D )   =>    |- [_ A / x ]_ C = [_ A / y ]_ D
Theorem	cbvcsb 3129	Change bound variables in a class substitution. Interestingly, this does not require any bound variable conditions on A. (Contributed by Jeff Hankins, 13-Sep-2009.) (Revised by Mario Carneiro, 11-Dec-2016.)
|- F/_ y C   &    |- F/_ x D   &    |- ( x = y -> C = D )   =>    |- [_ A / x ]_ C = [_ A / y ]_ D
Theorem	cbvcsbv 3130*	Change the bound variable of a proper substitution into a class using implicit substitution. (Contributed by NM, 30-Sep-2008.) (Revised by Mario Carneiro, 13-Oct-2016.)
|- ( x = y -> B = C )   =>    |- [_ A / x ]_ B = [_ A / y ]_ C
Theorem	csbeq1d 3131	Equality deduction for proper substitution into a class. (Contributed by NM, 3-Dec-2005.)
|- ( ph -> A = B )   =>    |- ( ph -> [_ A / x ]_ C = [_ B / x ]_ C )
Theorem	csbid 3132	Analog of sbid 1820 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
|- [_ x / x ]_ A = A
Theorem	csbeq1a 3133	Equality theorem for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
|- ( x = A -> B = [_ A / x ]_ B )
Theorem	csbco 3134*	Composition law for chained substitutions into a class. Use the weaker csbcow 3135 when possible. (Contributed by NM, 10-Nov-2005.) (New usage is discouraged.)
|- [_ A / y ]_ [_ y / x ]_ B = [_ A / x ]_ B
Theorem	csbcow 3135*	Composition law for chained substitutions into a class. Version of csbco 3134 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 10-Nov-2005.) (Revised by GG, 25-Aug-2024.)
|- [_ A / y ]_ [_ y / x ]_ B = [_ A / x ]_ B
Theorem	csbtt 3136	Substitution doesn't affect a constant B (in which x is not free). (Contributed by Mario Carneiro, 14-Oct-2016.)
|- ( ( A e. V /\ F/_ x B ) -> [_ A / x ]_ B = B )
Theorem	csbconstgf 3137	Substitution doesn't affect a constant B (in which x is not free). (Contributed by NM, 10-Nov-2005.)
|- F/_ x B   =>    |- ( A e. V -> [_ A / x ]_ B = B )
Theorem	csbconstg 3138*	Substitution doesn't affect a constant B (in which x is not free). csbconstgf 3137 with distinct variable requirement. (Contributed by Alan Sare, 22-Jul-2012.)
|- ( A e. V -> [_ A / x ]_ B = B )
Theorem	sbcel12g 3139	Distribute proper substitution through a membership relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- ( A e. V -> ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. [_ A / x ]_ C ) )
Theorem	sbceqg 3140	Distribute proper substitution through an equality relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- ( A e. V -> ( [. A / x ]. B = C <-> [_ A / x ]_ B = [_ A / x ]_ C ) )
Theorem	sbcnel12g 3141	Distribute proper substitution through negated membership. (Contributed by Andrew Salmon, 18-Jun-2011.)
|- ( A e. V -> ( [. A / x ]. B e/ C <-> [_ A / x ]_ B e/ [_ A / x ]_ C ) )
Theorem	sbcne12g 3142	Distribute proper substitution through an inequality. (Contributed by Andrew Salmon, 18-Jun-2011.)
|- ( A e. V -> ( [. A / x ]. B =/= C <-> [_ A / x ]_ B =/= [_ A / x ]_ C ) )
Theorem	sbcel1g 3143*	Move proper substitution in and out of a membership relation. Note that the scope of [. A / x ]. is the wff B e. C, whereas the scope of [_ A / x ]_ is the class B. (Contributed by NM, 10-Nov-2005.)
|- ( A e. V -> ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. C ) )
Theorem	sbceq1g 3144*	Move proper substitution to first argument of an equality. (Contributed by NM, 30-Nov-2005.)
|- ( A e. V -> ( [. A / x ]. B = C <-> [_ A / x ]_ B = C ) )
Theorem	sbcel2g 3145*	Move proper substitution in and out of a membership relation. (Contributed by NM, 14-Nov-2005.)
|- ( A e. V -> ( [. A / x ]. B e. C <-> B e. [_ A / x ]_ C ) )
Theorem	sbceq2g 3146*	Move proper substitution to second argument of an equality. (Contributed by NM, 30-Nov-2005.)
|- ( A e. V -> ( [. A / x ]. B = C <-> B = [_ A / x ]_ C ) )
Theorem	csbcomg 3147*	Commutative law for double substitution into a class. (Contributed by NM, 14-Nov-2005.)
|- ( ( A e. V /\ B e. W ) -> [_ A / x ]_ [_ B / y ]_ C = [_ B / y ]_ [_ A / x ]_ C )
Theorem	csbeq2 3148	Substituting into equivalent classes gives equivalent results. (Contributed by Giovanni Mascellani, 9-Apr-2018.)
|- ( A. x B = C -> [_ A / x ]_ B = [_ A / x ]_ C )
Theorem	csbeq2d 3149	Formula-building deduction for class substitution. (Contributed by NM, 22-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
|- F/ x ph   &    |- ( ph -> B = C )   =>    |- ( ph -> [_ A / x ]_ B = [_ A / x ]_ C )
Theorem	csbeq2dv 3150*	Formula-building deduction for class substitution. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
|- ( ph -> B = C )   =>    |- ( ph -> [_ A / x ]_ B = [_ A / x ]_ C )
Theorem	csbeq2i 3151	Formula-building inference for class substitution. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
|- B = C   =>    |- [_ A / x ]_ B = [_ A / x ]_ C
Theorem	csbvarg 3152	The proper substitution of a class for setvar variable results in the class (if the class exists). (Contributed by NM, 10-Nov-2005.)
|- ( A e. V -> [_ A / x ]_ x = A )
Theorem	sbccsbg 3153*	Substitution into a wff expressed in terms of substitution into a class. (Contributed by NM, 15-Aug-2007.)
|- ( A e. V -> ( [. A / x ]. ph <-> y e. [_ A / x ]_ { y | ph } ) )
Theorem	sbccsb2g 3154	Substitution into a wff expressed in using substitution into a class. (Contributed by NM, 27-Nov-2005.)
|- ( A e. V -> ( [. A / x ]. ph <-> A e. [_ A / x ]_ { x | ph } ) )
Theorem	nfcsb1d 3155	Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
|- ( ph -> F/_ x A )   =>    |- ( ph -> F/_ x [_ A / x ]_ B )
Theorem	nfcsb1 3156	Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
|- F/_ x A   =>    |- F/_ x [_ A / x ]_ B
Theorem	nfcsb1v 3157*	Bound-variable hypothesis builder for substitution into a class. (Contributed by NM, 17-Aug-2006.) (Revised by Mario Carneiro, 12-Oct-2016.)
|- F/_ x [_ A / x ]_ B
Theorem	nfsbcdw 3158*	Version of nfsbcd 3048 with a disjoint variable condition. (Contributed by NM, 23-Nov-2005.) (Revised by GG, 10-Jan-2024.)
|- F/ y ph   &    |- ( ph -> F/_ x A )   &    |- ( ph -> F/ x ps )   =>    |- ( ph -> F/ x [. A / y ]. ps )
Theorem	nfsbcw 3159*	Bound-variable hypothesis builder for class substitution. Version of nfsbc 3049 with a disjoint variable condition, which in the future may make it possible to reduce axiom usage. (Contributed by NM, 7-Sep-2014.) (Revised by GG, 10-Jan-2024.)
|- F/_ x A   &    |- F/ x ph   =>    |- F/ x [. A / y ]. ph
Theorem	nfcsbd 3160	Deduction version of nfcsb 3162. (Contributed by NM, 21-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.)
|- F/ y ph   &    |- ( ph -> F/_ x A )   &    |- ( ph -> F/_ x B )   =>    |- ( ph -> F/_ x [_ A / y ]_ B )
Theorem	nfcsbw 3161*	Bound-variable hypothesis builder for substitution into a class. Version of nfcsb 3162 with a disjoint variable condition. (Contributed by Mario Carneiro, 12-Oct-2016.) (Revised by GG, 10-Jan-2024.)
|- F/_ x A   &    |- F/_ x B   =>    |- F/_ x [_ A / y ]_ B
Theorem	nfcsb 3162	Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
|- F/_ x A   &    |- F/_ x B   =>    |- F/_ x [_ A / y ]_ B
Theorem	csbhypf 3163*	Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 2850 for class substitution version. (Contributed by NM, 19-Dec-2008.)
|- F/_ x A   &    |- F/_ x C   &    |- ( x = A -> B = C )   =>    |- ( y = A -> [_ y / x ]_ B = C )
Theorem	csbiebt 3164*	Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 3168.) (Contributed by NM, 11-Nov-2005.)
|- ( ( A e. V /\ F/_ x C ) -> ( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) )
Theorem	csbiedf 3165*	Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 13-Oct-2016.)
|- F/ x ph   &    |- ( ph -> F/_ x C )   &    |- ( ph -> A e. V )   &    |- ( ( ph /\ x = A ) -> B = C )   =>    |- ( ph -> [_ A / x ]_ B = C )
Theorem	csbieb 3166*	Bidirectional conversion between an implicit class substitution hypothesis x = A -> B = C and its explicit substitution equivalent. (Contributed by NM, 2-Mar-2008.)
|- A e. _V   &    |- F/_ x C   =>    |- ( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C )
Theorem	csbiebg 3167*	Bidirectional conversion between an implicit class substitution hypothesis x = A -> B = C and its explicit substitution equivalent. (Contributed by NM, 24-Mar-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)
|- F/_ x C   =>    |- ( A e. V -> ( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) )
Theorem	csbiegf 3168*	Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 11-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
|- ( A e. V -> F/_ x C )   &    |- ( x = A -> B = C )   =>    |- ( A e. V -> [_ A / x ]_ B = C )
Theorem	csbief 3169*	Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 26-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
|- A e. _V   &    |- F/_ x C   &    |- ( x = A -> B = C )   =>    |- [_ A / x ]_ B = C
Theorem	csbie 3170*	Conversion of implicit substitution to explicit substitution into a class. (Contributed by AV, 2-Dec-2019.)
|- A e. _V   &    |- ( x = A -> B = C )   =>    |- [_ A / x ]_ B = C
Theorem	csbied 3171*	Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Mario Carneiro, 13-Oct-2016.)
|- ( ph -> A e. V )   &    |- ( ( ph /\ x = A ) -> B = C )   =>    |- ( ph -> [_ A / x ]_ B = C )
Theorem	csbied2 3172*	Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by Mario Carneiro, 2-Jan-2017.)
|- ( ph -> A e. V )   &    |- ( ph -> A = B )   &    |- ( ( ph /\ x = B ) -> C = D )   =>    |- ( ph -> [_ A / x ]_ C = D )
Theorem	csbie2t 3173*	Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 3174). (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 13-Oct-2016.)
|- A e. _V   &    |- B e. _V   =>    |- ( A. x A. y ( ( x = A /\ y = B ) -> C = D ) -> [_ A / x ]_ [_ B / y ]_ C = D )
Theorem	csbie2 3174*	Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 27-Aug-2007.)
|- A e. _V   &    |- B e. _V   &    |- ( ( x = A /\ y = B ) -> C = D )   =>    |- [_ A / x ]_ [_ B / y ]_ C = D
Theorem	csbie2g 3175*	Conversion of implicit substitution to explicit class substitution. This version of sbcie 3063 avoids a disjointness condition on x and A by substituting twice. (Contributed by Mario Carneiro, 11-Nov-2016.)
|- ( x = y -> B = C )   &    |- ( y = A -> C = D )   =>    |- ( A e. V -> [_ A / x ]_ B = D )
Theorem	sbcnestgf 3176	Nest the composition of two substitutions. (Contributed by Mario Carneiro, 11-Nov-2016.)
|- ( ( A e. V /\ A. y F/ x ph ) -> ( [. A / x ]. [. B / y ]. ph <-> [. [_ A / x ]_ B / y ]. ph ) )
Theorem	csbnestgf 3177	Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)
|- ( ( A e. V /\ A. y F/_ x C ) -> [_ A / x ]_ [_ B / y ]_ C = [_ [_ A / x ]_ B / y ]_ C )
Theorem	sbcnestg 3178*	Nest the composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
|- ( A e. V -> ( [. A / x ]. [. B / y ]. ph <-> [. [_ A / x ]_ B / y ]. ph ) )
Theorem	csbnestg 3179*	Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)
|- ( A e. V -> [_ A / x ]_ [_ B / y ]_ C = [_ [_ A / x ]_ B / y ]_ C )
Theorem	csbnest1g 3180	Nest the composition of two substitutions. (Contributed by NM, 23-May-2006.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
|- ( A e. V -> [_ A / x ]_ [_ B / x ]_ C = [_ [_ A / x ]_ B / x ]_ C )
Theorem	csbidmg 3181*	Idempotent law for class substitutions. (Contributed by NM, 1-Mar-2008.)
|- ( A e. V -> [_ A / x ]_ [_ A / x ]_ B = [_ A / x ]_ B )
Theorem	sbcco3g 3182*	Composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)
|- ( x = A -> B = C )   =>    |- ( A e. V -> ( [. A / x ]. [. B / y ]. ph <-> [. C / y ]. ph ) )
Theorem	csbco3g 3183*	Composition of two class substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)
|- ( x = A -> B = C )   =>    |- ( A e. V -> [_ A / x ]_ [_ B / y ]_ D = [_ C / y ]_ D )
Theorem	rspcsbela 3184*	Special case related to rspsbc 3112. (Contributed by NM, 10-Dec-2005.) (Proof shortened by Eric Schmidt, 17-Jan-2007.)
|- ( ( A e. B /\ A. x e. B C e. D ) -> [_ A / x ]_ C e. D )
Theorem	sbnfc2 3185*	Two ways of expressing " x is (effectively) not free in A." (Contributed by Mario Carneiro, 14-Oct-2016.)
|- ( F/_ x A <-> A. y A. z [_ y / x ]_ A = [_ z / x ]_ A )
Theorem	csbabg 3186*	Move substitution into a class abstraction. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- ( A e. V -> [_ A / x ]_ { y | ph } = { y | [. A / x ]. ph } )
Theorem	cbvralcsf 3187	A more general version of cbvralf 2756 that doesn't require A and B to be distinct from x or y. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.)
|- F/_ y A   &    |- F/_ x B   &    |- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> A = B )   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( A. x e. A ph <-> A. y e. B ps )
Theorem	cbvrexcsf 3188	A more general version of cbvrexf 2757 that has no distinct variable restrictions. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.) (Proof shortened by Mario Carneiro, 7-Dec-2014.)
|- F/_ y A   &    |- F/_ x B   &    |- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> A = B )   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( E. x e. A ph <-> E. y e. B ps )
Theorem	cbvreucsf 3189	A more general version of cbvreuv 2767 that has no distinct variable rextrictions. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.)
|- F/_ y A   &    |- F/_ x B   &    |- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> A = B )   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( E! x e. A ph <-> E! y e. B ps )
Theorem	cbvrabcsf 3190	A more general version of cbvrab 2797 with no distinct variable restrictions. (Contributed by Andrew Salmon, 13-Jul-2011.)
|- F/_ y A   &    |- F/_ x B   &    |- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> A = B )   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- { x e. A | ph } = { y e. B | ps }
Theorem	cbvralv2 3191*	Rule used to change the bound variable in a restricted universal quantifier with implicit substitution which also changes the quantifier domain. (Contributed by David Moews, 1-May-2017.)
|- ( x = y -> ( ps <-> ch ) )   &    |- ( x = y -> A = B )   =>    |- ( A. x e. A ps <-> A. y e. B ch )
Theorem	cbvrexv2 3192*	Rule used to change the bound variable in a restricted existential quantifier with implicit substitution which also changes the quantifier domain. (Contributed by David Moews, 1-May-2017.)
|- ( x = y -> ( ps <-> ch ) )   &    |- ( x = y -> A = B )   =>    |- ( E. x e. A ps <-> E. y e. B ch )
Theorem	rspc2vd 3193*	Deduction version of 2-variable restricted specialization, using implicit substitution. Notice that the class D for the second set variable y may depend on the first set variable x. (Contributed by AV, 29-Mar-2021.)
|- ( x = A -> ( th <-> ch ) )   &    |- ( y = B -> ( ch <-> ps ) )   &    |- ( ph -> A e. C )   &    |- ( ( ph /\ x = A ) -> D = E )   &    |- ( ph -> B e. E )   =>    |- ( ph -> ( A. x e. C A. y e. D th -> ps ) )
2.1.11  Define basic set operations and relations
Syntax	cdif 3194	Extend class notation to include class difference (read: " A minus B").
class ( A \ B )
Syntax	cun 3195	Extend class notation to include union of two classes (read: " A union B").
class ( A u. B )
Syntax	cin 3196	Extend class notation to include the intersection of two classes (read: " A intersect B").
class ( A i^i B )
Syntax	wss 3197	Extend wff notation to include the subclass relation. This is read " A is a subclass of B " or " B includes A". When A exists as a set, it is also read " A is a subset of B".
wff A C_ B
Theorem	difjust 3198*	Soundness justification theorem for df-dif 3199. (Contributed by Rodolfo Medina, 27-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- { x | ( x e. A /\ -. x e. B ) } = { y | ( y e. A /\ -. y e. B ) }
Definition	df-dif 3199*	Define class difference, also called relative complement. Definition 5.12 of [TakeutiZaring] p. 20. Contrast this operation with union ( A u. B ) (df-un 3201) and intersection ( A i^i B ) (df-in 3203). Several notations are used in the literature; we chose the \ convention used in Definition 5.3 of [Eisenberg] p. 67 instead of the more common minus sign to reserve the latter for later use in, e.g., arithmetic. We will use the terminology " A excludes B " to mean A \ B. We will use " B is removed from A " to mean A \ { B } i.e. the removal of an element or equivalently the exclusion of a singleton. (Contributed by NM, 29-Apr-1994.)
|- ( A \ B ) = { x | ( x e. A /\ -. x e. B ) }
Theorem	unjust 3200*	Soundness justification theorem for df-un 3201. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- { x | ( x e. A \/ x e. B ) } = { y | ( y e. A \/ y e. B ) }