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Theorem breqd 3816
Description: Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
Hypothesis
Ref Expression
breq1d.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
breqd  |-  ( ph  ->  ( C A D  <-> 
C B D ) )

Proof of Theorem breqd
StepHypRef Expression
1 breq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 breq 3807 . 2  |-  ( A  =  B  ->  ( C A D  <->  C B D ) )
31, 2syl 14 1  |-  ( ph  ->  ( C A D  <-> 
C B D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1285   class class class wbr 3805
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 2065
This theorem depends on definitions:  df-bi 115  df-cleq 2076  df-clel 2079  df-br 3806
This theorem is referenced by:  breq123d  3819  sbcbr12g  3855  sprmpt2  5911  supeq123d  6498  shftfibg  9909  shftfib  9912  2shfti  9920
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