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Theorem breq12i 4032
Description: Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)
Hypotheses
Ref Expression
breq1i.1  |-  A  =  B
breq12i.2  |-  C  =  D
Assertion
Ref Expression
breq12i  |-  ( A R C  <->  B R D )

Proof of Theorem breq12i
StepHypRef Expression
1 breq1i.1 . 2  |-  A  =  B
2 breq12i.2 . 2  |-  C  =  D
3 breq12 4028 . 2  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A R C  <-> 
B R D ) )
41, 2, 3mp2an 653 1  |-  ( A R C  <->  B R D )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623   class class class wbr 4023
This theorem is referenced by:  3brtr3g  4054  3brtr4g  4055  caovord2  6032  domunfican  7129  ltsonq  8593  ltanq  8595  ltmnq  8596  prlem934  8657  prlem936  8671  ltsosr  8716  ltasr  8722  ltneg  9274  leneg  9277  inelr  9736  lt2sqi  11192  le2sqi  11193  nn0le2msqi  11282  mdsldmd1i  22911  axlowdimlem6  24575
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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