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Theorem breq123d 4168
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 4167 . 2  |-  ( ph  ->  ( A R C  <-> 
B R D ) )
4 breq123d.2 . . 3  |-  ( ph  ->  R  =  S )
54breqd 4165 . 2  |-  ( ph  ->  ( B R D  <-> 
B S D ) )
63, 5bitrd 245 1  |-  ( ph  ->  ( A R C  <-> 
B S D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649   class class class wbr 4154
This theorem is referenced by:  sbcbrg  4203  fmptco  5841  xpsle  13734  invfuc  14099  yonedainv  14306  fmptcof2  23919  subofld  24072  fnwe2val  26816  aomclem8  26829  iscvlat  29439  paddfval  29912  lhpset  30110  tendofset  30873  diaffval  31146
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-br 4155
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