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Theorem breq12i 3945
 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 3941 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝑅𝐶𝐵𝑅𝐷))
41, 2, 3mp2an 423 1 (𝐴𝑅𝐶𝐵𝑅𝐷)
 Colors of variables: wff set class Syntax hints:   ↔ wb 104   = wceq 1332   class class class wbr 3936 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3079  df-sn 3537  df-pr 3538  df-op 3540  df-br 3937 This theorem is referenced by:  3brtr3g  3968  3brtr4g  3969  caovord2  5950  ltneg  8247  leneg  8250  inelr  8369  lt2sqi  10410  le2sqi  10411
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