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Theorem breq123d 3943
Description: Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
Hypotheses
Ref Expression
breq1d.1  |-  ( ph  ->  A  =  B )
breq123d.2  |-  ( ph  ->  R  =  S )
breq123d.3  |-  ( ph  ->  C  =  D )
Assertion
Ref Expression
breq123d  |-  ( ph  ->  ( A R C  <-> 
B S D ) )

Proof of Theorem breq123d
StepHypRef Expression
1 breq1d.1 . . 3  |-  ( ph  ->  A  =  B )
2 breq123d.3 . . 3  |-  ( ph  ->  C  =  D )
31, 2breq12d 3942 . 2  |-  ( ph  ->  ( A R C  <-> 
B R D ) )
4 breq123d.2 . . 3  |-  ( ph  ->  R  =  S )
54breqd 3940 . 2  |-  ( ph  ->  ( B R D  <-> 
B S D ) )
63, 5bitrd 187 1  |-  ( ph  ->  ( A R C  <-> 
B S D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1331   class class class wbr 3929
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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930
This theorem is referenced by:  sbcbrg  3982  fmptco  5586
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