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Theorem List for Metamath Proof Explorer - 3001-3100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsbc2iegf 3001* 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 3002* 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 3003* 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 3004* 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 3005* Lemma for sbccom 3006. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 18-Oct-2016.)
 |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. B  /  y ]. [. A  /  x ].
 ph )
 
Theoremsbccom 3006* 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 3007* 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 3008* Interchange class substitution and restricted existential quantifier. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
 |-  ( ( A  e.  V  /\  F/_ y A ) 
 ->  ( [. A  /  x ]. E. y  e.  B  ph  <->  E. y  e.  B  [. A  /  x ]. ph ) )
 
Theoremsbcralg 3009* 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 ) )
 
Theoremsbcrexg 3010* Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. E. y  e.  B  ph  <->  E. y  e.  B  [. A  /  x ]. ph ) )
 
Theoremsbcreug 3011* Interchange class substitution and restricted uniqueness 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 3012* 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 ) )
 
Theoremra4sbc 3013* 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 1897 and a4sbc 2947. See also ra4sbca 3014 and ra4csbela 3082. (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 )
 )
 
Theoremra4sbca 3014* 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 )
 
Theoremra4esbca 3015* Existence form of ra4sbca 3014. (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 )
 
Theorema4esbc 3016 Existence form of a4sbc 2947. (Contributed by Mario Carneiro, 18-Nov-2016.)
 |-  ( [. A  /  x ]. ph  ->  E. x ph )
 
Theorema4esbcd 3017 form of a4sbc 2947. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ph  ->  [. A  /  x ]. ps )   =>    |-  ( ph  ->  E. x ps )
 
Theoremsbcth2 3018* 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 3019 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 1773. (Contributed by NM, 16-Jan-2004.)
 |- 
 F/ x ph   =>    |-  ( A. x  e.  A  ( ph  ->  ps )  ->  ( ph  ->  A. x  e.  A  ps ) )
 
Theoremrmo3 3020* Restricted "at most one" using explicit substitution. (Contributed by NM, 4-Nov-2012.)
 |- 
 F/ y ph   =>    |-  ( E* x ( x  e.  A  /\  ph )  <->  A. x  e.  A  A. y  e.  A  ( ( ph  /\  [
 y  /  x ] ph )  ->  x  =  y ) )
 
Theoremrmob 3021* Consequence of "at most one", using implicit substitution. (Contributed by NM, 2-Jan-2015.)
 |-  ( x  =  B  ->  ( ph  <->  ps ) )   &    |-  ( x  =  C  ->  (
 ph 
 <->  ch ) )   =>    |-  ( ( E* x ( x  e.  A  /\  ph )  /\  ( B  e.  A  /\  ps ) )  ->  ( B  =  C  <->  ( C  e.  A  /\  ch ) ) )
 
Theoremrmoi 3022* Consequence of "at most one", using implicit substitution. (Contributed by NM, 4-Nov-2012.)
 |-  ( x  =  B  ->  ( ph  <->  ps ) )   &    |-  ( x  =  C  ->  (
 ph 
 <->  ch ) )   =>    |-  ( ( E* x ( 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 3023 Extend class notation to include the proper substitution of a class for a set into another class.
 class  [_ A  /  x ]_ B
 
Definitiondf-csb 3024* 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 2935, to prevent ambiguity. Theorem sbcel1g 3042 shows an example of how ambiguity could arise if we didn't use distinguished brackets. Theorem sbccsbg 3051 recreates substitution into a wff from this definition. (Contributed by NM, 10-Nov-2005.)
 |-  [_ A  /  x ]_ B  =  { y  |  [. A  /  x ]. y  e.  B }
 
Theoremcsb2 3025* 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 3026 Analog of dfsbcq 2937 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
 |-  ( A  =  B  -> 
 [_ A  /  x ]_ C  =  [_ B  /  x ]_ C )
 
Theoremcbvcsb 3027 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 3028* 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 3029 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 3030 Analog of sbid 1896 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
 |-  [_ x  /  x ]_ A  =  A
 
Theoremcsbeq1a 3031 Equality theorem for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
 |-  ( x  =  A  ->  B  =  [_ A  /  x ]_ B )
 
Theoremcsbco 3032* Composition law for chained substitutions into a class. (Contributed by NM, 10-Nov-2005.)
 |-  [_ A  /  y ]_ [_ y  /  x ]_ B  =  [_ A  /  x ]_ B
 
Theoremcsbexg 3033 The existence of proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
 |-  ( ( A  e.  V  /\  A. x  B  e.  W )  ->  [_ A  /  x ]_ B  e.  _V )
 
Theoremcsbex 3034 The existence of proper substitution into a class. (Contributed by NM, 7-Aug-2007.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  [_ A  /  x ]_ B  e.  _V
 
Theoremcsbtt 3035 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 3036 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 3037* Substitution doesn't affect a constant  B (in which  x is not free). csbconstgf 3036 with distinct variable requirement. (Contributed by Alan Sare, 22-Jul-2012.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_ B  =  B )
 
Theoremsbcel12g 3038 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 3039 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 3040 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 3041 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 3042* 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 3043* 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 3044* 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 3045* 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 3046* 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 3047 Formula-building deduction rule 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 3048* Formula-building deduction rule 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 3049 Formula-building inference rule 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 3050 The proper substitution of a class for set variable results in the class (if the class exists). (Contributed by NM, 10-Nov-2005.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_ x  =  A )
 
Theoremsbccsbg 3051* 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 3052 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 3053 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 3054 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 3055* 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 3056 Deduction version of nfcsb 3057. (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 3057 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 3058* Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 2784 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 3059* Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 3063.) (Contributed by NM, 11-Nov-2005.)
 |-  ( ( A  e.  V  /\  F/_ x C ) 
 ->  ( A. x ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C ) )
 
Theoremcsbiedf 3060* 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 3061* 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 3062* 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 3063* 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 3064* 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
 
Theoremcsbied 3065* 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 3066* 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 3067* Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 3068). (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 3068* 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 3069* Conversion of implicit substitution to explicit class substitution. This version of sbcie 2969 avoids a disjointness condition on  x ,  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 3070 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 3071 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 3072* 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 3073* 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 )
 
TheoremcsbnestgOLD 3074* Nest the composition of two substitutions. (New usage is discouraged.) (Contributed by NM, 23-Nov-2005.)
 |-  ( ( A  e.  V  /\  A. x  B  e.  W )  ->  [_ A  /  x ]_ [_ B  /  y ]_ C  =  [_
 [_ A  /  x ]_ B  /  y ]_ C )
 
Theoremcsbnest1g 3075 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 )
 
Theoremcsbnest1gOLD 3076* Nest the composition of two substitutions. Obsolete as of 11-Nov-2016. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)
 |-  ( ( A  e.  V  /\  A. x  B  e.  W )  ->  [_ A  /  x ]_ [_ B  /  x ]_ C  =  [_
 [_ A  /  x ]_ B  /  x ]_ C )
 
Theoremcsbidmg 3077* Idempotent law for class substitutions. (Contributed by NM, 1-Mar-2008.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_
 [_ A  /  x ]_ B  =  [_ A  /  x ]_ B )
 
Theoremsbcco3g 3078* 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 ) )
 
Theoremsbcco3gOLD 3079* Composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (New usage is discouraged.)
 |-  ( x  =  A  ->  B  =  C )   =>    |-  ( ( A  e.  V  /\  A. x  B  e.  W )  ->  ( [. A  /  x ].
 [. B  /  y ]. ph  <->  [. C  /  y ]. ph ) )
 
Theoremcsbco3g 3080* 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 )
 
Theoremcsbco3gOLD 3081* Composition of two class substitutions. Obsolete as of 11-Nov-2016. (Contributed by NM, 27-Nov-2005.) (New usage is discouraged.)
 |-  ( x  =  A  ->  B  =  D )   =>    |-  ( ( A  e.  V  /\  A. x  B  e.  W )  ->  [_ A  /  x ]_ [_ B  /  y ]_ C  =  [_ D  /  y ]_ C )
 
Theoremra4csbela 3082* Special case related to ra4sbc 3013. (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 3083* 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 3084* 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 3085 A more general version of cbvralf 2712 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 3086 A more general version of cbvrexf 2713 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 3087 A more general version of cbvreuv 2719 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 3088 A more general version of cbvrab 2738 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 3089* 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 3090* 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 3091 Extend class notation to include class difference (read: " A minus  B").
 class  ( A  \  B )
 
Syntaxcun 3092 Extend class notation to include union of two classes (read: " A union  B").
 class  ( A  u.  B )
 
Syntaxcin 3093 Extend class notation to include the intersection of two classes (read: " A intersect  B").
 class  ( A  i^i  B )
 
Syntaxwss 3094 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
 
Syntaxwpss 3095 Extend wff notation with proper subclass relation.
 wff  A  C.  B
 
Theoremdifjust 3096* Soundness justification theorem for df-dif 3097. (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 3097* Define class difference, also called relative complement. Definition 5.12 of [TakeutiZaring] p. 20. For example,  ( { 1 ,  3 }  \  { 1 ,  8 } )  =  {
3 } (ex-dif 20718). Contrast this operation with union  ( A  u.  B
) (df-un 3099) and intersection  ( A  i^i  B ) (df-in 3101). 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 3098* Soundness justification theorem for df-un 3099. (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 ) }
 
Definitiondf-un 3099* Define the union of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example,  ( { 1 ,  3 }  u.  {
1 ,  8 } )  =  { 1 ,  3 ,  8 } (ex-un 20719). Contrast this operation with difference  ( A  \  B ) (df-dif 3097) and intersection  ( A  i^i  B ) (df-in 3101). For an alternate definition in terms of class difference, requiring no dummy variables, see dfun2 3346. For union defined in terms of intersection, see dfun3 3349. (Contributed by NM, 23-Aug-1993.)
 |-  ( A  u.  B )  =  { x  |  ( x  e.  A  \/  x  e.  B ) }
 
Theoreminjust 3100* Soundness justification theorem for df-in 3101. (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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