Theorem breq12i 5107

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	⊢ (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷)

Proof of Theorem breq12i

Step	Hyp	Ref	Expression
1		breq1i.1	. 2 ⊢ 𝐴 = 𝐵
2		breq12i.2	. 2 ⊢ 𝐶 = 𝐷
3		breq12 5103	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷))
4	1, 2, 3	mp2an 692	1 ⊢ (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷)

Syntax hints:   ↔ wb 206   = wceq 1541   class class class wbr 5098

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099

This theorem is referenced by:  3brtr3g  5131  3brtr4g  5132  caovord2  7570  domunfican  9222  ltsonq  10880  ltanq  10882  ltmnq  10883  prlem934  10944  prlem936  10958  ltsosr  11005  ltasr  11011  ltneg  11637  leneg  11640  lt2sqi  14112  le2sqi  14113  nn0le2msqi  14190  2sqreuop  27429  2sqreuopnn  27430  2sqreuoplt  27431  2sqreuopltb  27432  2sqreuopnnlt  27433  2sqreuopnnltb  27434  axlowdimlem6  29020  upgrwlkcompim  29716  clwlkcompbp  29855  mdsldmd1i  32406  fldext2chn  33885  constrextdg2lem  33905  divcnvlin  35927  ditgeq123i  36403  cbvditgvw2  36443  relowlpssretop  37569  2ap1caineq  42399  fsumlessf  45823  climlimsupcex  46013  liminfltlimsupex  46025  liminflelimsupcex  46041  sge0xaddlem2  46678  eubrdm  47282  isgrlim2  48229  iscmgmALT  48470  iscsgrpALT  48472