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Theorem breq123d 4218
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 4217 . 2  |-  ( ph  ->  ( A R C  <-> 
B R D ) )
4 breq123d.2 . . 3  |-  ( ph  ->  R  =  S )
54breqd 4215 . 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 1652   class class class wbr 4204
This theorem is referenced by:  sbcbrg  4253  fmptco  5893  xpsle  13798  invfuc  14163  yonedainv  14370  fmptcof2  24068  subofld  24237  inftmrel  24242  isinftm  24243  fnwe2val  27105  aomclem8  27117  iscvlat  30048  paddfval  30521  lhpset  30719  tendofset  31482  diaffval  31755
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205
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