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Theorem breq12i 4221
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 4217 . 2  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A R C  <-> 
B R D ) )
41, 2, 3mp2an 654 1  |-  ( A R C  <->  B R D )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1652   class class class wbr 4212
This theorem is referenced by:  3brtr3g  4243  3brtr4g  4244  caovord2  6259  domunfican  7379  ltsonq  8846  ltanq  8848  ltmnq  8849  prlem934  8910  prlem936  8924  ltsosr  8969  ltasr  8975  ltneg  9528  leneg  9531  inelr  9990  lt2sqi  11470  le2sqi  11471  nn0le2msqi  11560  mdsldmd1i  23834  divcnvlin  25212  axlowdimlem6  25886
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 2417
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 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213
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