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Theorem breq12i 3884
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 (𝐴𝑅𝐶𝐵𝑅𝐷)

Proof of Theorem breq12i
StepHypRef Expression
1 breq1i.1 . 2 𝐴 = 𝐵
2 breq12i.2 . 2 𝐶 = 𝐷
3 breq12 3880 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝑅𝐶𝐵𝑅𝐷))
41, 2, 3mp2an 420 1 (𝐴𝑅𝐶𝐵𝑅𝐷)
Colors of variables: wff set class
Syntax hints:  wb 104   = wceq 1299   class class class wbr 3875
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 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-un 3025  df-sn 3480  df-pr 3481  df-op 3483  df-br 3876
This theorem is referenced by:  3brtr3g  3906  3brtr4g  3907  caovord2  5875  ltneg  8091  leneg  8094  inelr  8212  lt2sqi  10221  le2sqi  10222
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