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Theorem breq123d 4037
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 4036 . 2  |-  ( ph  ->  ( A R C  <-> 
B R D ) )
4 breq123d.2 . . 3  |-  ( ph  ->  R  =  S )
54breqd 4034 . 2  |-  ( ph  ->  ( B R D  <-> 
B S D ) )
63, 5bitrd 244 1  |-  ( ph  ->  ( A R C  <-> 
B S D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623   class class class wbr 4023
This theorem is referenced by:  sbcbrg  4072  fmptco  5691  xpsle  13483  invfuc  13848  yonedainv  14055  fmptcof2  23229  fnwe2val  27146  aomclem8  27159  iscvlat  29513  paddfval  29986  lhpset  30184  tendofset  30947  diaffval  31220
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024
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