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Theorem breqd 3910
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 3901 . 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 104    = wceq 1316   class class class wbr 3899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-17 1491  ax-ial 1499  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-cleq 2110  df-clel 2113  df-br 3900
This theorem is referenced by:  breq123d  3913  breqdi  3914  sbcbr12g  3953  supeq123d  6846  shftfibg  10560  shftfib  10563  2shfti  10571  lmbr  12309
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