Theorem breq12i 5111

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 5107	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷))
4	1, 2, 3	mp2an 705	1 ⊢ (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1569   class class class wbr 5102

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103

This theorem is used by:  3brtr3g  5137  3brtr4g  5138  caovord2  7633  domunfican  9314  ltsonq  11055  ltanq  11057  ltmnq  11058  prlem934  11119  prlem936  11133  ltsosr  11180  ltasr  11186  ltneg  11817  leneg  11820  lt2sqi  14333  le2sqi  14334  nn0le2msqi  14412  2sqreuop  27790  2sqreuopnn  27791  2sqreuoplt  27792  2sqreuopltb  27793  2sqreuopnnlt  27794  2sqreuopnnltb  27795  axlowdimlem6  29526  upgrwlkcompim  30224  clwlkcompbp  30370  mdsldmd1i  32934  fldext2chn  34403  constrextdg2lem  34423  divcnvlin  36573  ditgeq123i  37073  cbvditgvw2  37113  relowlpssretop  38363  2ap1caineq  43271  fsumlessf  46665  climlimsupcex  46855  liminfltlimsupex  46867  liminflelimsupcex  46883  sge0xaddlem2  47520  eubrdm  48182  isgrlim2  49157  iscmgmALT  49397  iscsgrpALT  49399