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Theorem List for Intuitionistic Logic Explorer - 3101-3200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdifeq2 3101 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 3102 Equality theorem for class difference. (Contributed by FL, 31-Aug-2009.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  \  C )  =  ( B  \  D ) )
 
Theoremdifeq1i 3103 Inference adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)
 |-  A  =  B   =>    |-  ( A  \  C )  =  ( B  \  C )
 
Theoremdifeq2i 3104 Inference adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)
 |-  A  =  B   =>    |-  ( C  \  A )  =  ( C  \  B )
 
Theoremdifeq12i 3105 Equality inference for class difference. (Contributed by NM, 29-Aug-2004.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A  \  C )  =  ( B  \  D )
 
Theoremdifeq1d 3106 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 3107 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 3108 Equality deduction for class difference. (Contributed by FL, 29-May-2014.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A  \  C )  =  ( B  \  D ) )
 
Theoremdifeqri 3109* 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 3110 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 )
 
Theoremeldifi 3111 Implication of membership in a class difference. (Contributed by NM, 29-Apr-1994.)
 |-  ( A  e.  ( B  \  C )  ->  A  e.  B )
 
Theoremeldifn 3112 Implication of membership in a class difference. (Contributed by NM, 3-May-1994.)
 |-  ( A  e.  ( B  \  C )  ->  -.  A  e.  C )
 
Theoremelndif 3113 A set does not belong to a class excluding it. (Contributed by NM, 27-Jun-1994.)
 |-  ( A  e.  B  ->  -.  A  e.  ( C  \  B ) )
 
Theoremdifdif 3114 Double class difference. Exercise 11 of [TakeutiZaring] p. 22. (Contributed by NM, 17-May-1998.)
 |-  ( A  \  ( B  \  A ) )  =  A
 
Theoremdifss 3115 Subclass relationship for class difference. Exercise 14 of [TakeutiZaring] p. 22. (Contributed by NM, 29-Apr-1994.)
 |-  ( A  \  B )  C_  A
 
Theoremdifssd 3116 A difference of two classes is contained in the minuend. Deduction form of difss 3115. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  ( A  \  B )  C_  A )
 
Theoremdifss2 3117 If a class is contained in a difference, it is contained in the minuend. (Contributed by David Moews, 1-May-2017.)
 |-  ( A  C_  ( B  \  C )  ->  A  C_  B )
 
Theoremdifss2d 3118 If a class is contained in a difference, it is contained in the minuend. Deduction form of difss2 3117. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C_  ( B  \  C ) )   =>    |-  ( ph  ->  A  C_  B )
 
Theoremssdifss 3119 Preservation of a subclass relationship by class difference. (Contributed by NM, 15-Feb-2007.)
 |-  ( A  C_  B  ->  ( A  \  C )  C_  B )
 
Theoremddifnel 3120* Double complement under universal class. The hypothesis corresponds to stability of membership in 
A, which is weaker than decidability (see dcimpstab 788). Actually, the conclusion is a characterization of stability of membership in a class (see ddifstab 3121) . Exercise 4.10(s) of [Mendelson] p. 231, but with an additional hypothesis. For a version without a hypothesis, but which only states that  A is a subset of  _V  \  ( _V  \  A ), see ddifss 3226. (Contributed by Jim Kingdon, 21-Jul-2018.)
 |-  ( -.  x  e.  ( _V  \  A )  ->  x  e.  A )   =>    |-  ( _V  \  ( _V  \  A ) )  =  A
 
Theoremddifstab 3121* A class is equal to its double complement if and only if it is stable (that is, membership in it is a stable property). (Contributed by BJ, 12-Dec-2021.)
 |-  ( ( _V  \  ( _V  \  A ) )  =  A  <->  A. xSTAB  x  e.  A )
 
Theoremssconb 3122 Contraposition law for subsets. (Contributed by NM, 22-Mar-1998.)
 |-  ( ( A  C_  C  /\  B  C_  C )  ->  ( A  C_  ( C  \  B )  <->  B  C_  ( C  \  A ) ) )
 
Theoremsscon 3123 Contraposition law for subsets. Exercise 15 of [TakeutiZaring] p. 22. (Contributed by NM, 22-Mar-1998.)
 |-  ( A  C_  B  ->  ( C  \  B )  C_  ( C  \  A ) )
 
Theoremssdif 3124 Difference law for subsets. (Contributed by NM, 28-May-1998.)
 |-  ( A  C_  B  ->  ( A  \  C )  C_  ( B  \  C ) )
 
Theoremssdifd 3125 If  A is contained in  B, then  ( A 
\  C ) is contained in  ( B  \  C ). Deduction form of ssdif 3124. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C_  B )   =>    |-  ( ph  ->  ( A  \  C )  C_  ( B  \  C ) )
 
Theoremsscond 3126 If  A is contained in  B, then  ( C 
\  B ) is contained in  ( C  \  A ). Deduction form of sscon 3123. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C_  B )   =>    |-  ( ph  ->  ( C  \  B )  C_  ( C  \  A ) )
 
Theoremssdifssd 3127 If  A is contained in  B, then  ( A 
\  C ) is also contained in  B. Deduction form of ssdifss 3119. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C_  B )   =>    |-  ( ph  ->  ( A  \  C )  C_  B )
 
Theoremssdif2d 3128 If  A is contained in  B and  C is contained in  D, then  ( A  \  D ) is contained in  ( B  \  C ). Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  C 
 C_  D )   =>    |-  ( ph  ->  ( A  \  D ) 
 C_  ( B  \  C ) )
 
Theoremraldifb 3129 Restricted universal quantification on a class difference in terms of an implication. (Contributed by Alexander van der Vekens, 3-Jan-2018.)
 |-  ( A. x  e.  A  ( x  e/  B  ->  ph )  <->  A. x  e.  ( A  \  B ) ph )
 
2.1.13.2  The union of two classes
 
Theoremelun 3130 Expansion of membership in class union. Theorem 12 of [Suppes] p. 25. (Contributed by NM, 7-Aug-1994.)
 |-  ( A  e.  ( B  u.  C )  <->  ( A  e.  B  \/  A  e.  C ) )
 
Theoremuneqri 3131* Inference from membership to union. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( x  e.  A  \/  x  e.  B )  <->  x  e.  C )   =>    |-  ( A  u.  B )  =  C
 
Theoremunidm 3132 Idempotent law for union of classes. Theorem 23 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  u.  A )  =  A
 
Theoremuncom 3133 Commutative law for union of classes. Exercise 6 of [TakeutiZaring] p. 17. (Contributed by NM, 25-Jun-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( A  u.  B )  =  ( B  u.  A )
 
Theoremequncom 3134 If a class equals the union of two other classes, then it equals the union of those two classes commuted. (Contributed by Alan Sare, 18-Feb-2012.)
 |-  ( A  =  ( B  u.  C )  <->  A  =  ( C  u.  B ) )
 
Theoremequncomi 3135 Inference form of equncom 3134. (Contributed by Alan Sare, 18-Feb-2012.)
 |-  A  =  ( B  u.  C )   =>    |-  A  =  ( C  u.  B )
 
Theoremuneq1 3136 Equality theorem for union of two classes. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  =  B  ->  ( A  u.  C )  =  ( B  u.  C ) )
 
Theoremuneq2 3137 Equality theorem for the union of two classes. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  =  B  ->  ( C  u.  A )  =  ( C  u.  B ) )
 
Theoremuneq12 3138 Equality theorem for union of two classes. (Contributed by NM, 29-Mar-1998.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  u.  C )  =  ( B  u.  D ) )
 
Theoremuneq1i 3139 Inference adding union to the right in a class equality. (Contributed by NM, 30-Aug-1993.)
 |-  A  =  B   =>    |-  ( A  u.  C )  =  ( B  u.  C )
 
Theoremuneq2i 3140 Inference adding union to the left in a class equality. (Contributed by NM, 30-Aug-1993.)
 |-  A  =  B   =>    |-  ( C  u.  A )  =  ( C  u.  B )
 
Theoremuneq12i 3141 Equality inference for union of two classes. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A  u.  C )  =  ( B  u.  D )
 
Theoremuneq1d 3142 Deduction adding union to the right in a class equality. (Contributed by NM, 29-Mar-1998.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  u.  C )  =  ( B  u.  C ) )
 
Theoremuneq2d 3143 Deduction adding union to the left in a class equality. (Contributed by NM, 29-Mar-1998.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  u.  A )  =  ( C  u.  B ) )
 
Theoremuneq12d 3144 Equality deduction for union of two classes. (Contributed by NM, 29-Sep-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A  u.  C )  =  ( B  u.  D ) )
 
Theoremnfun 3145 Bound-variable hypothesis builder for the union of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x ( A  u.  B )
 
Theoremunass 3146 Associative law for union of classes. Exercise 8 of [TakeutiZaring] p. 17. (Contributed by NM, 3-May-1994.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( ( A  u.  B )  u.  C )  =  ( A  u.  ( B  u.  C ) )
 
Theoremun12 3147 A rearrangement of union. (Contributed by NM, 12-Aug-2004.)
 |-  ( A  u.  ( B  u.  C ) )  =  ( B  u.  ( A  u.  C ) )
 
Theoremun23 3148 A rearrangement of union. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( ( A  u.  B )  u.  C )  =  ( ( A  u.  C )  u.  B )
 
Theoremun4 3149 A rearrangement of the union of 4 classes. (Contributed by NM, 12-Aug-2004.)
 |-  ( ( A  u.  B )  u.  ( C  u.  D ) )  =  ( ( A  u.  C )  u.  ( B  u.  D ) )
 
Theoremunundi 3150 Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
 |-  ( A  u.  ( B  u.  C ) )  =  ( ( A  u.  B )  u.  ( A  u.  C ) )
 
Theoremunundir 3151 Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
 |-  ( ( A  u.  B )  u.  C )  =  ( ( A  u.  C )  u.  ( B  u.  C ) )
 
Theoremssun1 3152 Subclass relationship for union of classes. Theorem 25 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)
 |-  A  C_  ( A  u.  B )
 
Theoremssun2 3153 Subclass relationship for union of classes. (Contributed by NM, 30-Aug-1993.)
 |-  A  C_  ( B  u.  A )
 
Theoremssun3 3154 Subclass law for union of classes. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  C_  B  ->  A  C_  ( B  u.  C ) )
 
Theoremssun4 3155 Subclass law for union of classes. (Contributed by NM, 14-Aug-1994.)
 |-  ( A  C_  B  ->  A  C_  ( C  u.  B ) )
 
Theoremelun1 3156 Membership law for union of classes. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  e.  B  ->  A  e.  ( B  u.  C ) )
 
Theoremelun2 3157 Membership law for union of classes. (Contributed by NM, 30-Aug-1993.)
 |-  ( A  e.  B  ->  A  e.  ( C  u.  B ) )
 
Theoremunss1 3158 Subclass law for union of classes. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( A  C_  B  ->  ( A  u.  C )  C_  ( B  u.  C ) )
 
Theoremssequn1 3159 A relationship between subclass and union. Theorem 26 of [Suppes] p. 27. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( A  C_  B  <->  ( A  u.  B )  =  B )
 
Theoremunss2 3160 Subclass law for union of classes. Exercise 7 of [TakeutiZaring] p. 18. (Contributed by NM, 14-Oct-1999.)
 |-  ( A  C_  B  ->  ( C  u.  A )  C_  ( C  u.  B ) )
 
Theoremunss12 3161 Subclass law for union of classes. (Contributed by NM, 2-Jun-2004.)
 |-  ( ( A  C_  B  /\  C  C_  D )  ->  ( A  u.  C )  C_  ( B  u.  D ) )
 
Theoremssequn2 3162 A relationship between subclass and union. (Contributed by NM, 13-Jun-1994.)
 |-  ( A  C_  B  <->  ( B  u.  A )  =  B )
 
Theoremunss 3163 The union of two subclasses is a subclass. Theorem 27 of [Suppes] p. 27 and its converse. (Contributed by NM, 11-Jun-2004.)
 |-  ( ( A  C_  C  /\  B  C_  C ) 
 <->  ( A  u.  B )  C_  C )
 
Theoremunssi 3164 An inference showing the union of two subclasses is a subclass. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  A  C_  C   &    |-  B  C_  C   =>    |-  ( A  u.  B )  C_  C
 
Theoremunssd 3165 A deduction showing the union of two subclasses is a subclass. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ph  ->  A  C_  C )   &    |-  ( ph  ->  B 
 C_  C )   =>    |-  ( ph  ->  ( A  u.  B ) 
 C_  C )
 
Theoremunssad 3166 If  ( A  u.  B ) is contained in  C, so is  A. One-way deduction form of unss 3163. Partial converse of unssd 3165. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  ( A  u.  B )  C_  C )   =>    |-  ( ph  ->  A  C_  C )
 
Theoremunssbd 3167 If  ( A  u.  B ) is contained in  C, so is  B. One-way deduction form of unss 3163. Partial converse of unssd 3165. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  ( A  u.  B )  C_  C )   =>    |-  ( ph  ->  B  C_  C )
 
Theoremssun 3168 A condition that implies inclusion in the union of two classes. (Contributed by NM, 23-Nov-2003.)
 |-  ( ( A  C_  B  \/  A  C_  C )  ->  A  C_  ( B  u.  C ) )
 
Theoremrexun 3169 Restricted existential quantification over union. (Contributed by Jeff Madsen, 5-Jan-2011.)
 |-  ( E. x  e.  ( A  u.  B ) ph  <->  ( E. x  e.  A  ph  \/  E. x  e.  B  ph ) )
 
Theoremralunb 3170 Restricted quantification over a union. (Contributed by Scott Fenton, 12-Apr-2011.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  ( A. x  e.  ( A  u.  B ) ph  <->  ( A. x  e.  A  ph  /\  A. x  e.  B  ph ) )
 
Theoremralun 3171 Restricted quantification over union. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ( A. x  e.  A  ph  /\  A. x  e.  B  ph )  ->  A. x  e.  ( A  u.  B ) ph )
 
2.1.13.3  The intersection of two classes
 
Theoremelin 3172 Expansion of membership in an intersection of two classes. Theorem 12 of [Suppes] p. 25. (Contributed by NM, 29-Apr-1994.)
 |-  ( A  e.  ( B  i^i  C )  <->  ( A  e.  B  /\  A  e.  C ) )
 
Theoremelini 3173 Membership in an intersection of two classes. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  A  e.  B   &    |-  A  e.  C   =>    |-  A  e.  ( B  i^i  C )
 
Theoremelind 3174 Deduce membership in an intersection of two classes. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ph  ->  X  e.  A )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  X  e.  ( A  i^i  B ) )
 
Theoremelinel1 3175 Membership in an intersection implies membership in the first set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( A  e.  ( B  i^i  C )  ->  A  e.  B )
 
Theoremelinel2 3176 Membership in an intersection implies membership in the second set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( A  e.  ( B  i^i  C )  ->  A  e.  C )
 
Theoremelin2 3177 Membership in a class defined as an intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  X  =  ( B  i^i  C )   =>    |-  ( A  e.  X 
 <->  ( A  e.  B  /\  A  e.  C ) )
 
Theoremelin1d 3178 Elementhood in the first set of an intersection - deduction version. (Contributed by Thierry Arnoux, 3-May-2020.)
 |-  ( ph  ->  X  e.  ( A  i^i  B ) )   =>    |-  ( ph  ->  X  e.  A )
 
Theoremelin2d 3179 Elementhood in the first set of an intersection - deduction version. (Contributed by Thierry Arnoux, 3-May-2020.)
 |-  ( ph  ->  X  e.  ( A  i^i  B ) )   =>    |-  ( ph  ->  X  e.  B )
 
Theoremelin3 3180 Membership in a class defined as a ternary intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  X  =  ( ( B  i^i  C )  i^i  D )   =>    |-  ( A  e.  X 
 <->  ( A  e.  B  /\  A  e.  C  /\  A  e.  D )
 )
 
Theoremincom 3181 Commutative law for intersection of classes. Exercise 7 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  i^i  B )  =  ( B  i^i  A )
 
Theoremineqri 3182* Inference from membership to intersection. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( x  e.  A  /\  x  e.  B )  <->  x  e.  C )   =>    |-  ( A  i^i  B )  =  C
 
Theoremineq1 3183 Equality theorem for intersection of two classes. (Contributed by NM, 14-Dec-1993.)
 |-  ( A  =  B  ->  ( A  i^i  C )  =  ( B  i^i  C ) )
 
Theoremineq2 3184 Equality theorem for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
 |-  ( A  =  B  ->  ( C  i^i  A )  =  ( C  i^i  B ) )
 
Theoremineq12 3185 Equality theorem for intersection of two classes. (Contributed by NM, 8-May-1994.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  i^i  C )  =  ( B  i^i  D ) )
 
Theoremineq1i 3186 Equality inference for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
 |-  A  =  B   =>    |-  ( A  i^i  C )  =  ( B  i^i  C )
 
Theoremineq2i 3187 Equality inference for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
 |-  A  =  B   =>    |-  ( C  i^i  A )  =  ( C  i^i  B )
 
Theoremineq12i 3188 Equality inference for intersection of two classes. (Contributed by NM, 24-Jun-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A  i^i  C )  =  ( B  i^i  D )
 
Theoremineq1d 3189 Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  i^i  C )  =  ( B  i^i  C ) )
 
Theoremineq2d 3190 Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  i^i  A )  =  ( C  i^i  B ) )
 
Theoremineq12d 3191 Equality deduction for intersection of two classes. (Contributed by NM, 24-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A  i^i  C )  =  ( B  i^i  D ) )
 
Theoremineqan12d 3192 Equality deduction for intersection of two classes. (Contributed by NM, 7-Feb-2007.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ps  ->  C  =  D )   =>    |-  ( ( ph  /\ 
 ps )  ->  ( A  i^i  C )  =  ( B  i^i  D ) )
 
Theoremdfss1 3193 A frequently-used variant of subclass definition df-ss 3001. (Contributed by NM, 10-Jan-2015.)
 |-  ( A  C_  B  <->  ( B  i^i  A )  =  A )
 
Theoremdfss5 3194 Another definition of subclasshood. Similar to df-ss 3001, dfss 3002, and dfss1 3193. (Contributed by David Moews, 1-May-2017.)
 |-  ( A  C_  B  <->  A  =  ( B  i^i  A ) )
 
Theoremnfin 3195 Bound-variable hypothesis builder for the intersection of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x ( A  i^i  B )
 
Theoremcsbing 3196 Distribute proper substitution through an intersection relation. (Contributed by Alan Sare, 22-Jul-2012.)
 |-  ( A  e.  B  -> 
 [_ A  /  x ]_ ( C  i^i  D )  =  ( [_ A  /  x ]_ C  i^i  [_ A  /  x ]_ D ) )
 
Theoremrabbi2dva 3197* Deduction from a wff to a restricted class abstraction. (Contributed by NM, 14-Jan-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  ( x  e.  B  <->  ps ) )   =>    |-  ( ph  ->  ( A  i^i  B )  =  { x  e.  A  |  ps }
 )
 
Theoreminidm 3198 Idempotent law for intersection of classes. Theorem 15 of [Suppes] p. 26. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  i^i  A )  =  A
 
Theoreminass 3199 Associative law for intersection of classes. Exercise 9 of [TakeutiZaring] p. 17. (Contributed by NM, 3-May-1994.)
 |-  ( ( A  i^i  B )  i^i  C )  =  ( A  i^i  ( B  i^i  C ) )
 
Theoremin12 3200 A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.)
 |-  ( A  i^i  ( B  i^i  C ) )  =  ( B  i^i  ( A  i^i  C ) )
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