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Theorem breq12i 4027
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 4023 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝑅𝐶𝐵𝑅𝐷))
41, 2, 3mp2an 426 1 (𝐴𝑅𝐶𝐵𝑅𝐷)
Colors of variables: wff set class
Syntax hints:  wb 105   = wceq 1364   class class class wbr 4018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019
This theorem is referenced by:  3brtr3g  4051  3brtr4g  4052  caovord2  6070  ltneg  8450  leneg  8453  inelr  8572  lt2sqi  10642  le2sqi  10643
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