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Theorem breq12i 4813
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 4809 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝑅𝐶𝐵𝑅𝐷))
41, 2, 3mp2an 710 1 (𝐴𝑅𝐶𝐵𝑅𝐷)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1632   class class class wbr 4804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805
This theorem is referenced by:  3brtr3g  4837  3brtr4g  4838  caovord2  7011  domunfican  8398  ltsonq  9983  ltanq  9985  ltmnq  9986  prlem934  10047  prlem936  10061  ltsosr  10107  ltasr  10113  ltneg  10720  leneg  10723  inelr  11202  lt2sqi  13146  le2sqi  13147  nn0le2msqi  13248  axlowdimlem6  26026  upgrwlkcompim  26749  clwlkcompbp  26888  mdsldmd1i  29499  divcnvlin  31925  relowlpssretop  33523  fsumlessf  40312  climlimsupcex  40504  liminfltlimsupex  40516  liminflelimsupcex  40532  sge0xaddlem2  41154  iscmgmALT  42370  iscsgrpALT  42372
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