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Theorem breq12i 4223
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 4219 . 2  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A R C  <-> 
B R D ) )
41, 2, 3mp2an 655 1  |-  ( A R C  <->  B R D )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1653   class class class wbr 4214
This theorem is referenced by:  3brtr3g  4245  3brtr4g  4246  caovord2  6261  domunfican  7381  ltsonq  8848  ltanq  8850  ltmnq  8851  prlem934  8912  prlem936  8926  ltsosr  8971  ltasr  8977  ltneg  9530  leneg  9533  inelr  9992  lt2sqi  11472  le2sqi  11473  nn0le2msqi  11562  mdsldmd1i  23836  divcnvlin  25214  axlowdimlem6  25888
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215
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