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Theorem List for Intuitionistic Logic Explorer - 2901-3000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsbcel21v 2901* Class substitution into a membership relation. One direction of sbcel2gv 2900 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)
 |-  ( [. B  /  x ]. A  e.  x  ->  A  e.  B )
 
Theoremsbcimdv 2902* Substitution analogue of Theorem 19.20 of [Margaris] p. 90 (alim 1391). (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 ) )
 
Theoremsbctt 2903 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 ) )
 
Theoremsbcgf 2904 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 ) )
 
Theoremsbc19.21g 2905 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 ) ) )
 
Theoremsbcg 2906* Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 2904. (Contributed by Alan Sare, 10-Nov-2012.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ph ) )
 
Theoremsbc2iegf 2907* 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 ) )
 
Theoremsbc2ie 2908* 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 )
 
Theoremsbc2iedv 2909* 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 ) )
 
Theoremsbc3ie 2910* 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 )
 
Theoremsbccomlem 2911* Lemma for sbccom 2912. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 18-Oct-2016.)
 |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. B  /  y ]. [. A  /  x ].
 ph )
 
Theoremsbccom 2912* 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 )
 
Theoremsbcralt 2913* 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 ) )
 
Theoremsbcrext 2914* 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 )
 )
 
Theoremsbcralg 2915* 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 ) )
 
Theoremsbcrex 2916* 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 )
 
Theoremsbcreug 2917* 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 ) )
 
Theoremsbcabel 2918* 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 ) )
 
Theoremrspsbc 2919* 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 1705 and spsbc 2849. See also rspsbca 2920 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 )
 )
 
Theoremrspsbca 2920* 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 )
 
Theoremrspesbca 2921* Existence form of rspsbca 2920. (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 )
 
Theoremspesbc 2922 Existence form of spsbc 2849. (Contributed by Mario Carneiro, 18-Nov-2016.)
 |-  ( [. A  /  x ]. ph  ->  E. x ph )
 
Theoremspesbcd 2923 form of spsbc 2849. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ph  ->  [. A  /  x ]. ps )   =>    |-  ( ph  ->  E. x ps )
 
Theoremsbcth2 2924* 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 )
 
Theoremra5 2925 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 1521. (Contributed by NM, 16-Jan-2004.)
 |- 
 F/ x ph   =>    |-  ( A. x  e.  A  ( ph  ->  ps )  ->  ( ph  ->  A. x  e.  A  ps ) )
 
Theoremrmo2ilem 2926* 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 )
 
Theoremrmo2i 2927* 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 )
 
Theoremrmo3 2928* 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 ) )
 
Theoremrmob 2929* 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 ) ) )
 
Theoremrmoi 2930* 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
 
Syntaxcsb 2931 Extend class notation to include the proper substitution of a class for a set into another class.
 class  [_ A  /  x ]_ B
 
Definitiondf-csb 2932* 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 2838, to prevent ambiguity. Theorem sbcel1g 2948 shows an example of how ambiguity could arise if we didn't use distinguished brackets. Theorem sbccsbg 2957 recreates substitution into a wff from this definition. (Contributed by NM, 10-Nov-2005.)
 |-  [_ A  /  x ]_ B  =  { y  |  [. A  /  x ]. y  e.  B }
 
Theoremcsb2 2933* 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 ) }
 
Theoremcsbeq1 2934 Analog of dfsbcq 2840 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
 |-  ( A  =  B  -> 
 [_ A  /  x ]_ C  =  [_ B  /  x ]_ C )
 
Theoremcbvcsb 2935 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
 
Theoremcbvcsbv 2936* 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
 
Theoremcsbeq1d 2937 Equality deduction for proper substitution into a class. (Contributed by NM, 3-Dec-2005.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  [_ A  /  x ]_ C  =  [_ B  /  x ]_ C )
 
Theoremcsbid 2938 Analog of sbid 1704 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
 |-  [_ x  /  x ]_ A  =  A
 
Theoremcsbeq1a 2939 Equality theorem for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
 |-  ( x  =  A  ->  B  =  [_ A  /  x ]_ B )
 
Theoremcsbco 2940* Composition law for chained substitutions into a class. (Contributed by NM, 10-Nov-2005.)
 |-  [_ A  /  y ]_ [_ y  /  x ]_ B  =  [_ A  /  x ]_ B
 
Theoremcsbtt 2941 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 )
 
Theoremcsbconstgf 2942 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 )
 
Theoremcsbconstg 2943* Substitution doesn't affect a constant  B (in which  x is not free). csbconstgf 2942 with distinct variable requirement. (Contributed by Alan Sare, 22-Jul-2012.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_ B  =  B )
 
Theoremsbcel12g 2944 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 ) )
 
Theoremsbceqg 2945 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 ) )
 
Theoremsbcnel12g 2946 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 ) )
 
Theoremsbcne12g 2947 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 ) )
 
Theoremsbcel1g 2948* 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 )
 )
 
Theoremsbceq1g 2949* 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 )
 )
 
Theoremsbcel2g 2950* 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 ) )
 
Theoremsbceq2g 2951* 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 ) )
 
Theoremcsbcomg 2952* 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 )
 
Theoremcsbeq2d 2953 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 )
 
Theoremcsbeq2dv 2954* 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 )
 
Theoremcsbeq2i 2955 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
 
Theoremcsbvarg 2956 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 )
 
Theoremsbccsbg 2957* 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 } ) )
 
Theoremsbccsb2g 2958 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 } ) )
 
Theoremnfcsb1d 2959 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 )
 
Theoremnfcsb1 2960 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
 |-  F/_ x A   =>    |-  F/_ x [_ A  /  x ]_ B
 
Theoremnfcsb1v 2961* 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
 
Theoremnfcsbd 2962 Deduction version of nfcsb 2963. (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 )
 
Theoremnfcsb 2963 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
 
Theoremcsbhypf 2964* Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 2668 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 )
 
Theoremcsbiebt 2965* Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 2969.) (Contributed by NM, 11-Nov-2005.)
 |-  ( ( A  e.  V  /\  F/_ x C ) 
 ->  ( A. x ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C ) )
 
Theoremcsbiedf 2966* 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 )
 
Theoremcsbieb 2967* 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 )
 
Theoremcsbiebg 2968* 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 ) )
 
Theoremcsbiegf 2969* 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 )
 
Theoremcsbief 2970* 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
 
Theoremcsbie 2971* 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
 
Theoremcsbied 2972* 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 )
 
Theoremcsbied2 2973* 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 )
 
Theoremcsbie2t 2974* Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 2975). (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 )
 
Theoremcsbie2 2975* 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
 
Theoremcsbie2g 2976* Conversion of implicit substitution to explicit class substitution. This version of sbcie 2871 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 )
 
Theoremsbcnestgf 2977 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 ) )
 
Theoremcsbnestgf 2978 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 )
 
Theoremsbcnestg 2979* 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 ) )
 
Theoremcsbnestg 2980* 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 )
 
Theoremcsbnest1g 2981 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 )
 
Theoremcsbidmg 2982* Idempotent law for class substitutions. (Contributed by NM, 1-Mar-2008.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_
 [_ A  /  x ]_ B  =  [_ A  /  x ]_ B )
 
Theoremsbcco3g 2983* 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 ) )
 
Theoremcsbco3g 2984* 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 )
 
Theoremrspcsbela 2985* Special case related to rspsbc 2919. (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 )
 
Theoremsbnfc2 2986* 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 )
 
Theoremcsbabg 2987* 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 } )
 
Theoremcbvralcsf 2988 A more general version of cbvralf 2584 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 )
 
Theoremcbvrexcsf 2989 A more general version of cbvrexf 2585 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 )
 
Theoremcbvreucsf 2990 A more general version of cbvreuv 2592 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 )
 
Theoremcbvrabcsf 2991 A more general version of cbvrab 2617 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 }
 
Theoremcbvralv2 2992* 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 )
 
Theoremcbvrexv2 2993* 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 )
 
2.1.11  Define basic set operations and relations
 
Syntaxcdif 2994 Extend class notation to include class difference (read: " A minus  B").
 class  ( A  \  B )
 
Syntaxcun 2995 Extend class notation to include union of two classes (read: " A union  B").
 class  ( A  u.  B )
 
Syntaxcin 2996 Extend class notation to include the intersection of two classes (read: " A intersect  B").
 class  ( A  i^i  B )
 
Syntaxwss 2997 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
 
Theoremdifjust 2998* Soundness justification theorem for df-dif 2999. (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 ) }
 
Definitiondf-dif 2999* Define class difference, also called relative complement. Definition 5.12 of [TakeutiZaring] p. 20. Contrast this operation with union  ( A  u.  B ) (df-un 3001) and intersection  ( A  i^i  B ) (df-in 3003). 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 ) }
 
Theoremunjust 3000* Soundness justification theorem for df-un 3001. (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 ) }
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