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Theorem List for Intuitionistic Logic Explorer - 3101-3200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsseldd 3101 Membership inference from subclass relationship. (Contributed by NM, 14-Dec-2004.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  C  e.  A )   =>    |-  ( ph  ->  C  e.  B )
 
Theoremssneld 3102 If a class is not in another class, it is also not in a subclass of that class. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C_  B )   =>    |-  ( ph  ->  ( -.  C  e.  B  ->  -.  C  e.  A ) )
 
Theoremssneldd 3103 If an element is not in a class, it is also not in a subclass of that class. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  -.  C  e.  B )   =>    |-  ( ph  ->  -.  C  e.  A )
 
Theoremssriv 3104* Inference based on subclass definition. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  e.  A  ->  x  e.  B )   =>    |-  A  C_  B
 
Theoremssrd 3105 Deduction based on subclass definition. (Contributed by Thierry Arnoux, 8-Mar-2017.)
 |- 
 F/ x ph   &    |-  F/_ x A   &    |-  F/_ x B   &    |-  ( ph  ->  ( x  e.  A  ->  x  e.  B ) )   =>    |-  ( ph  ->  A  C_  B )
 
Theoremssrdv 3106* Deduction based on subclass definition. (Contributed by NM, 15-Nov-1995.)
 |-  ( ph  ->  ( x  e.  A  ->  x  e.  B ) )   =>    |-  ( ph  ->  A  C_  B )
 
Theoremsstr2 3107 Transitivity of subclasses. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
 |-  ( A  C_  B  ->  ( B  C_  C  ->  A  C_  C )
 )
 
Theoremsstr 3108 Transitivity of subclasses. Theorem 6 of [Suppes] p. 23. (Contributed by NM, 5-Sep-2003.)
 |-  ( ( A  C_  B  /\  B  C_  C )  ->  A  C_  C )
 
Theoremsstri 3109 Subclass transitivity inference. (Contributed by NM, 5-May-2000.)
 |-  A  C_  B   &    |-  B  C_  C   =>    |-  A  C_  C
 
Theoremsstrd 3110 Subclass transitivity deduction. (Contributed by NM, 2-Jun-2004.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  B 
 C_  C )   =>    |-  ( ph  ->  A 
 C_  C )
 
Theoremsstrid 3111 Subclass transitivity deduction. (Contributed by NM, 6-Feb-2014.)
 |-  A  C_  B   &    |-  ( ph  ->  B 
 C_  C )   =>    |-  ( ph  ->  A 
 C_  C )
 
Theoremsstrdi 3112 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ph  ->  A  C_  B )   &    |-  B  C_  C   =>    |-  ( ph  ->  A  C_  C )
 
Theoremsylan9ss 3113 A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ps  ->  B 
 C_  C )   =>    |-  ( ( ph  /\ 
 ps )  ->  A  C_  C )
 
Theoremsylan9ssr 3114 A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ps  ->  B 
 C_  C )   =>    |-  ( ( ps 
 /\  ph )  ->  A  C_  C )
 
Theoremeqss 3115 The subclass relationship is antisymmetric. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
 
Theoremeqssi 3116 Infer equality from two subclass relationships. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 9-Sep-1993.)
 |-  A  C_  B   &    |-  B  C_  A   =>    |-  A  =  B
 
Theoremeqssd 3117 Equality deduction from two subclass relationships. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 27-Jun-2004.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  B 
 C_  A )   =>    |-  ( ph  ->  A  =  B )
 
Theoremeqrd 3118 Deduce equality of classes from equivalence of membership. (Contributed by Thierry Arnoux, 21-Mar-2017.)
 |- 
 F/ x ph   &    |-  F/_ x A   &    |-  F/_ x B   &    |-  ( ph  ->  ( x  e.  A  <->  x  e.  B ) )   =>    |-  ( ph  ->  A  =  B )
 
Theoremeqelssd 3119* Equality deduction from subclass relationship and membership. (Contributed by AV, 21-Aug-2022.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  x  e.  A )   =>    |-  ( ph  ->  A  =  B )
 
Theoremssid 3120 Any class is a subclass of itself. Exercise 10 of [TakeutiZaring] p. 18. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
 |-  A  C_  A
 
Theoremssidd 3121 Weakening of ssid 3120. (Contributed by BJ, 1-Sep-2022.)
 |-  ( ph  ->  A  C_  A )
 
Theoremssv 3122 Any class is a subclass of the universal class. (Contributed by NM, 31-Oct-1995.)
 |-  A  C_  _V
 
Theoremsseq1 3123 Equality theorem for subclasses. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
 |-  ( A  =  B  ->  ( A  C_  C  <->  B 
 C_  C ) )
 
Theoremsseq2 3124 Equality theorem for the subclass relationship. (Contributed by NM, 25-Jun-1998.)
 |-  ( A  =  B  ->  ( C  C_  A  <->  C 
 C_  B ) )
 
Theoremsseq12 3125 Equality theorem for the subclass relationship. (Contributed by NM, 31-May-1999.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  C_  C 
 <->  B  C_  D )
 )
 
Theoremsseq1i 3126 An equality inference for the subclass relationship. (Contributed by NM, 18-Aug-1993.)
 |-  A  =  B   =>    |-  ( A  C_  C 
 <->  B  C_  C )
 
Theoremsseq2i 3127 An equality inference for the subclass relationship. (Contributed by NM, 30-Aug-1993.)
 |-  A  =  B   =>    |-  ( C  C_  A 
 <->  C  C_  B )
 
Theoremsseq12i 3128 An equality inference for the subclass relationship. (Contributed by NM, 31-May-1999.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A  C_  C  <->  B 
 C_  D )
 
Theoremsseq1d 3129 An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  C_  C  <->  B  C_  C ) )
 
Theoremsseq2d 3130 An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  C_  A  <->  C  C_  B ) )
 
Theoremsseq12d 3131 An equality deduction for the subclass relationship. (Contributed by NM, 31-May-1999.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A  C_  C  <->  B  C_  D ) )
 
Theoremeqsstri 3132 Substitution of equality into a subclass relationship. (Contributed by NM, 16-Jul-1995.)
 |-  A  =  B   &    |-  B  C_  C   =>    |-  A  C_  C
 
Theoremeqsstrri 3133 Substitution of equality into a subclass relationship. (Contributed by NM, 19-Oct-1999.)
 |-  B  =  A   &    |-  B  C_  C   =>    |-  A  C_  C
 
Theoremsseqtri 3134 Substitution of equality into a subclass relationship. (Contributed by NM, 28-Jul-1995.)
 |-  A  C_  B   &    |-  B  =  C   =>    |-  A  C_  C
 
Theoremsseqtrri 3135 Substitution of equality into a subclass relationship. (Contributed by NM, 4-Apr-1995.)
 |-  A  C_  B   &    |-  C  =  B   =>    |-  A  C_  C
 
Theoremeqsstrd 3136 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  B 
 C_  C )   =>    |-  ( ph  ->  A 
 C_  C )
 
Theoremeqsstrrd 3137 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
 |-  ( ph  ->  B  =  A )   &    |-  ( ph  ->  B 
 C_  C )   =>    |-  ( ph  ->  A 
 C_  C )
 
Theoremsseqtrd 3138 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  A 
 C_  C )
 
Theoremsseqtrrd 3139 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  C  =  B )   =>    |-  ( ph  ->  A 
 C_  C )
 
Theorem3sstr3i 3140 Substitution of equality in both sides of a subclass relationship. (Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
 |-  A  C_  B   &    |-  A  =  C   &    |-  B  =  D   =>    |-  C  C_  D
 
Theorem3sstr4i 3141 Substitution of equality in both sides of a subclass relationship. (Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
 |-  A  C_  B   &    |-  C  =  A   &    |-  D  =  B   =>    |-  C  C_  D
 
Theorem3sstr3g 3142 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)
 |-  ( ph  ->  A  C_  B )   &    |-  A  =  C   &    |-  B  =  D   =>    |-  ( ph  ->  C  C_  D )
 
Theorem3sstr4g 3143 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
 |-  ( ph  ->  A  C_  B )   &    |-  C  =  A   &    |-  D  =  B   =>    |-  ( ph  ->  C  C_  D )
 
Theorem3sstr3d 3144 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  A  =  C )   &    |-  ( ph  ->  B  =  D )   =>    |-  ( ph  ->  C  C_  D )
 
Theorem3sstr4d 3145 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 30-Nov-1995.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  C  =  A )   &    |-  ( ph  ->  D  =  B )   =>    |-  ( ph  ->  C  C_  D )
 
Theoremeqsstrid 3146 B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
 |-  A  =  B   &    |-  ( ph  ->  B  C_  C )   =>    |-  ( ph  ->  A  C_  C )
 
Theoremeqsstrrid 3147 B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
 |-  B  =  A   &    |-  ( ph  ->  B  C_  C )   =>    |-  ( ph  ->  A  C_  C )
 
Theoremsseqtrdi 3148 A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
 |-  ( ph  ->  A  C_  B )   &    |-  B  =  C   =>    |-  ( ph  ->  A  C_  C )
 
Theoremsseqtrrdi 3149 A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
 |-  ( ph  ->  A  C_  B )   &    |-  C  =  B   =>    |-  ( ph  ->  A  C_  C )
 
Theoremsseqtrid 3150 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  B  C_  A   &    |-  ( ph  ->  A  =  C )   =>    |-  ( ph  ->  B 
 C_  C )
 
Theoremsseqtrrid 3151 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  B  C_  A   &    |-  ( ph  ->  C  =  A )   =>    |-  ( ph  ->  B 
 C_  C )
 
Theoremeqsstrdi 3152 A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  A  =  B )   &    |-  B  C_  C   =>    |-  ( ph  ->  A  C_  C )
 
Theoremeqsstrrdi 3153 A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  B  =  A )   &    |-  B  C_  C   =>    |-  ( ph  ->  A  C_  C )
 
Theoremeqimss 3154 Equality implies the subclass relation. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
 |-  ( A  =  B  ->  A  C_  B )
 
Theoremeqimss2 3155 Equality implies the subclass relation. (Contributed by NM, 23-Nov-2003.)
 |-  ( B  =  A  ->  A  C_  B )
 
Theoremeqimssi 3156 Infer subclass relationship from equality. (Contributed by NM, 6-Jan-2007.)
 |-  A  =  B   =>    |-  A  C_  B
 
Theoremeqimss2i 3157 Infer subclass relationship from equality. (Contributed by NM, 7-Jan-2007.)
 |-  A  =  B   =>    |-  B  C_  A
 
Theoremnssne1 3158 Two classes are different if they don't include the same class. (Contributed by NM, 23-Apr-2015.)
 |-  ( ( A  C_  B  /\  -.  A  C_  C )  ->  B  =/=  C )
 
Theoremnssne2 3159 Two classes are different if they are not subclasses of the same class. (Contributed by NM, 23-Apr-2015.)
 |-  ( ( A  C_  C  /\  -.  B  C_  C )  ->  A  =/=  B )
 
Theoremnssr 3160* Negation of subclass relationship. One direction of Exercise 13 of [TakeutiZaring] p. 18. (Contributed by Jim Kingdon, 15-Jul-2018.)
 |-  ( E. x ( x  e.  A  /\  -.  x  e.  B ) 
 ->  -.  A  C_  B )
 
Theoremnelss 3161 Demonstrate by witnesses that two classes lack a subclass relation. (Contributed by Stefan O'Rear, 5-Feb-2015.)
 |-  ( ( A  e.  B  /\  -.  A  e.  C )  ->  -.  B  C_  C )
 
Theoremssrexf 3162 Restricted existential quantification follows from a subclass relationship. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  ( A  C_  B  ->  ( E. x  e.  A  ph  ->  E. x  e.  B  ph ) )
 
Theoremssrmof 3163 "At most one" existential quantification restricted to a subclass. (Contributed by Thierry Arnoux, 8-Oct-2017.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  ( A  C_  B  ->  ( E* x  e.  B  ph  ->  E* x  e.  A  ph ) )
 
Theoremssralv 3164* Quantification restricted to a subclass. (Contributed by NM, 11-Mar-2006.)
 |-  ( A  C_  B  ->  ( A. x  e.  B  ph  ->  A. x  e.  A  ph ) )
 
Theoremssrexv 3165* Existential quantification restricted to a subclass. (Contributed by NM, 11-Jan-2007.)
 |-  ( A  C_  B  ->  ( E. x  e.  A  ph  ->  E. x  e.  B  ph ) )
 
Theoremralss 3166* Restricted universal quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( A  C_  B  ->  ( A. x  e.  A  ph  <->  A. x  e.  B  ( x  e.  A  -> 
 ph ) ) )
 
Theoremrexss 3167* Restricted existential quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( A  C_  B  ->  ( E. x  e.  A  ph  <->  E. x  e.  B  ( x  e.  A  /\  ph ) ) )
 
Theoremss2ab 3168 Class abstractions in a subclass relationship. (Contributed by NM, 3-Jul-1994.)
 |-  ( { x  |  ph
 }  C_  { x  |  ps }  <->  A. x ( ph  ->  ps ) )
 
Theoremabss 3169* Class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)
 |-  ( { x  |  ph
 }  C_  A  <->  A. x ( ph  ->  x  e.  A ) )
 
Theoremssab 3170* Subclass of a class abstraction. (Contributed by NM, 16-Aug-2006.)
 |-  ( A  C_  { x  |  ph }  <->  A. x ( x  e.  A  ->  ph )
 )
 
Theoremssabral 3171* The relation for a subclass of a class abstraction is equivalent to restricted quantification. (Contributed by NM, 6-Sep-2006.)
 |-  ( A  C_  { x  |  ph }  <->  A. x  e.  A  ph )
 
Theoremss2abi 3172 Inference of abstraction subclass from implication. (Contributed by NM, 31-Mar-1995.)
 |-  ( ph  ->  ps )   =>    |-  { x  |  ph }  C_  { x  |  ps }
 
Theoremss2abdv 3173* Deduction of abstraction subclass from implication. (Contributed by NM, 29-Jul-2011.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  { x  |  ps }  C_ 
 { x  |  ch } )
 
Theoremabssdv 3174* Deduction of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)
 |-  ( ph  ->  ( ps  ->  x  e.  A ) )   =>    |-  ( ph  ->  { x  |  ps }  C_  A )
 
Theoremabssi 3175* Inference of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)
 |-  ( ph  ->  x  e.  A )   =>    |- 
 { x  |  ph } 
 C_  A
 
Theoremss2rab 3176 Restricted abstraction classes in a subclass relationship. (Contributed by NM, 30-May-1999.)
 |-  ( { x  e.  A  |  ph }  C_  { x  e.  A  |  ps }  <->  A. x  e.  A  ( ph  ->  ps )
 )
 
Theoremrabss 3177* Restricted class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)
 |-  ( { x  e.  A  |  ph }  C_  B 
 <-> 
 A. x  e.  A  ( ph  ->  x  e.  B ) )
 
Theoremssrab 3178* Subclass of a restricted class abstraction. (Contributed by NM, 16-Aug-2006.)
 |-  ( B  C_  { x  e.  A  |  ph }  <->  ( B  C_  A  /\  A. x  e.  B  ph ) )
 
Theoremssrabdv 3179* Subclass of a restricted class abstraction (deduction form). (Contributed by NM, 31-Aug-2006.)
 |-  ( ph  ->  B  C_  A )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  ps )   =>    |-  ( ph  ->  B  C_ 
 { x  e.  A  |  ps } )
 
Theoremrabssdv 3180* Subclass of a restricted class abstraction (deduction form). (Contributed by NM, 2-Feb-2015.)
 |-  ( ( ph  /\  x  e.  A  /\  ps )  ->  x  e.  B )   =>    |-  ( ph  ->  { x  e.  A  |  ps }  C_  B )
 
Theoremss2rabdv 3181* Deduction of restricted abstraction subclass from implication. (Contributed by NM, 30-May-2006.)
 |-  ( ( ph  /\  x  e.  A )  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  { x  e.  A  |  ps }  C_  { x  e.  A  |  ch }
 )
 
Theoremss2rabi 3182 Inference of restricted abstraction subclass from implication. (Contributed by NM, 14-Oct-1999.)
 |-  ( x  e.  A  ->  ( ph  ->  ps )
 )   =>    |- 
 { x  e.  A  |  ph }  C_  { x  e.  A  |  ps }
 
Theoremrabss2 3183* Subclass law for restricted abstraction. (Contributed by NM, 18-Dec-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( A  C_  B  ->  { x  e.  A  |  ph }  C_  { x  e.  B  |  ph } )
 
Theoremssab2 3184* Subclass relation for the restriction of a class abstraction. (Contributed by NM, 31-Mar-1995.)
 |- 
 { x  |  ( x  e.  A  /\  ph ) }  C_  A
 
Theoremssrab2 3185* Subclass relation for a restricted class. (Contributed by NM, 19-Mar-1997.)
 |- 
 { x  e.  A  |  ph }  C_  A
 
Theoremssrabeq 3186* If the restricting class of a restricted class abstraction is a subset of this restricted class abstraction, it is equal to this restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.)
 |-  ( V  C_  { x  e.  V  |  ph }  <->  V  =  { x  e.  V  |  ph
 } )
 
Theoremrabssab 3187 A restricted class is a subclass of the corresponding unrestricted class. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |- 
 { x  e.  A  |  ph }  C_  { x  |  ph }
 
Theoremuniiunlem 3188* A subset relationship useful for converting union to indexed union using dfiun2 or dfiun2g and intersection to indexed intersection using dfiin2 . (Contributed by NM, 5-Oct-2006.) (Proof shortened by Mario Carneiro, 26-Sep-2015.)
 |-  ( A. x  e.  A  B  e.  D  ->  ( A. x  e.  A  B  e.  C  <->  { y  |  E. x  e.  A  y  =  B }  C_  C ) )
 
2.1.13  The difference, union, and intersection of two classes
 
2.1.13.1  The difference of two classes
 
Theoremdfdif3 3189* Alternate definition of class difference. Definition of relative set complement in Section 2.3 of [Pierik], p. 10. (Contributed by BJ and Jim Kingdon, 16-Jun-2022.)
 |-  ( A  \  B )  =  { x  e.  A  |  A. y  e.  B  x  =/=  y }
 
Theoremdifeq1 3190 Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( A  =  B  ->  ( A  \  C )  =  ( B  \  C ) )
 
Theoremdifeq2 3191 Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( A  =  B  ->  ( C  \  A )  =  ( C  \  B ) )
 
Theoremdifeq12 3192 Equality theorem for class difference. (Contributed by FL, 31-Aug-2009.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  \  C )  =  ( B  \  D ) )
 
Theoremdifeq1i 3193 Inference adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)
 |-  A  =  B   =>    |-  ( A  \  C )  =  ( B  \  C )
 
Theoremdifeq2i 3194 Inference adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)
 |-  A  =  B   =>    |-  ( C  \  A )  =  ( C  \  B )
 
Theoremdifeq12i 3195 Equality inference for class difference. (Contributed by NM, 29-Aug-2004.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A  \  C )  =  ( B  \  D )
 
Theoremdifeq1d 3196 Deduction adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  \  C )  =  ( B  \  C ) )
 
Theoremdifeq2d 3197 Deduction adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  \  A )  =  ( C  \  B ) )
 
Theoremdifeq12d 3198 Equality deduction for class difference. (Contributed by FL, 29-May-2014.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A  \  C )  =  ( B  \  D ) )
 
Theoremdifeqri 3199* Inference from membership to difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( ( x  e.  A  /\  -.  x  e.  B )  <->  x  e.  C )   =>    |-  ( A  \  B )  =  C
 
Theoremnfdif 3200 Bound-variable hypothesis builder for class difference. (Contributed by NM, 3-Dec-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x ( A 
 \  B )
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