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Theorem breq 3795
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 2143 . 2  |-  ( R  =  S  ->  ( <. A ,  B >.  e.  R  <->  <. A ,  B >.  e.  S ) )
2 df-br 3794 . 2  |-  ( A R B  <->  <. A ,  B >.  e.  R )
3 df-br 3794 . 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 1285    e. wcel 1434   <.cop 3409   class class class wbr 3793
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 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-cleq 2075  df-clel 2078  df-br 3794
This theorem is referenced by:  breqi  3799  breqd  3804  poeq1  4062  soeq1  4078  frforeq1  4106  weeq1  4119  fveq1  5208  foeqcnvco  5461  f1eqcocnv  5462  isoeq2  5473  isoeq3  5474  ofreq  5746  supeq3  6462  shftfvalg  9844  shftfval  9847
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