Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  breq12i Structured version   Visualization version   GIF version

Theorem breq12i 5074
 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 5070 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝑅𝐶𝐵𝑅𝐷))
41, 2, 3mp2an 690 1 (𝐴𝑅𝐶𝐵𝑅𝐷)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 208   = wceq 1533   class class class wbr 5065 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-br 5066 This theorem is referenced by:  3brtr3g  5098  3brtr4g  5099  caovord2  7359  domunfican  8790  ltsonq  10390  ltanq  10392  ltmnq  10393  prlem934  10454  prlem936  10468  ltsosr  10515  ltasr  10521  ltneg  11139  leneg  11142  inelr  11627  lt2sqi  13551  le2sqi  13552  nn0le2msqi  13626  2sqreuop  26037  2sqreuopnn  26038  2sqreuoplt  26039  2sqreuopltb  26040  2sqreuopnnlt  26041  2sqreuopnnltb  26042  axlowdimlem6  26732  upgrwlkcompim  27423  clwlkcompbp  27562  mdsldmd1i  30107  divcnvlin  32964  relowlpssretop  34644  fsumlessf  41858  climlimsupcex  42050  liminfltlimsupex  42062  liminflelimsupcex  42078  sge0xaddlem2  42717  eubrdm  43272  iscmgmALT  44132  iscsgrpALT  44134
 Copyright terms: Public domain W3C validator