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Theorem breq123d 4453
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 4452 . 2  |-  ( ph  ->  ( A R C  <-> 
B R D ) )
4 breq123d.2 . . 3  |-  ( ph  ->  R  =  S )
54breqd 4450 . 2  |-  ( ph  ->  ( B R D  <-> 
B S D ) )
63, 5bitrd 253 1  |-  ( ph  ->  ( A R C  <-> 
B S D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1398   class class class wbr 4439
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440
This theorem is referenced by:  sbcbr123  4490  fmptco  6040  xpsle  15073  invfuc  15465  yonedainv  15752  opphllem3  24325  lmif  24355  islmib  24357  fmptcof2  27727  submomnd  27937  sgnsv  27954  inftmrel  27961  isinftm  27962  submarchi  27967  suborng  28043  fnwe2val  31237  aomclem8  31249  iscvlat  35464  paddfval  35937  lhpset  36135  tendofset  36900  diaffval  37173
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