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Theorem breq12i 3876
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 3872 . 2  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A R C  <-> 
B R D ) )
41, 2, 3mp2an 418 1  |-  ( A R C  <->  B R D )
Colors of variables: wff set class
Syntax hints:    <-> wb 104    = wceq 1296   class class class wbr 3867
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-un 3017  df-sn 3472  df-pr 3473  df-op 3475  df-br 3868
This theorem is referenced by:  3brtr3g  3898  3brtr4g  3899  caovord2  5855  ltneg  8037  leneg  8040  inelr  8158  lt2sqi  10173  le2sqi  10174
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