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Theorem breq 3847
Description: Equality theorem for binary relations. (Contributed by NM, 4-Jun-1995.)
Assertion
Ref Expression
breq  |-  ( R  =  S  ->  ( A R B  <->  A S B ) )

Proof of Theorem breq
StepHypRef Expression
1 eleq2 2151 . 2  |-  ( R  =  S  ->  ( <. A ,  B >.  e.  R  <->  <. A ,  B >.  e.  S ) )
2 df-br 3846 . 2  |-  ( A R B  <->  <. A ,  B >.  e.  R )
3 df-br 3846 . 2  |-  ( A S B  <->  <. A ,  B >.  e.  S )
41, 2, 33bitr4g 221 1  |-  ( R  =  S  ->  ( A R B  <->  A S B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1289    e. wcel 1438   <.cop 3449   class class class wbr 3845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-cleq 2081  df-clel 2084  df-br 3846
This theorem is referenced by:  breqi  3851  breqd  3856  poeq1  4126  soeq1  4142  frforeq1  4170  weeq1  4183  fveq1  5304  foeqcnvco  5569  f1eqcocnv  5570  isoeq2  5581  isoeq3  5582  ofreq  5859  supeq3  6683  shftfvalg  10248  shftfval  10251
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