Theorem breq12i 5151

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 5147	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷))
4	1, 2, 3	mp2an 692	1 ⊢ (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷)

Syntax hints:   ↔ wb 206   = wceq 1539   class class class wbr 5142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143

This theorem is referenced by:  3brtr3g  5175  3brtr4g  5176  caovord2  7646  domunfican  9362  ltsonq  11010  ltanq  11012  ltmnq  11013  prlem934  11074  prlem936  11088  ltsosr  11135  ltasr  11141  ltneg  11764  leneg  11767  inelr  12257  lt2sqi  14229  le2sqi  14230  nn0le2msqi  14307  2sqreuop  27507  2sqreuopnn  27508  2sqreuoplt  27509  2sqreuopltb  27510  2sqreuopnnlt  27511  2sqreuopnnltb  27512  axlowdimlem6  28963  upgrwlkcompim  29662  clwlkcompbp  29803  mdsldmd1i  32351  fldext2chn  33770  constrextdg2lem  33790  divcnvlin  35734  ditgeq123i  36211  cbvditgvw2  36251  relowlpssretop  37366  2ap1caineq  42147  fsumlessf  45597  climlimsupcex  45789  liminfltlimsupex  45801  liminflelimsupcex  45817  sge0xaddlem2  46454  eubrdm  47053  isgrlim2  47955  iscmgmALT  48145  iscsgrpALT  48147