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  9313  ltsonq  11054  ltanq  11056  ltmnq  11057  prlem934  11118  prlem936  11132  ltsosr  11179  ltasr  11185  ltneg  11816  leneg  11819  lt2sqi  14332  le2sqi  14333  nn0le2msqi  14411  2sqreuop  27789  2sqreuopnn  27790  2sqreuoplt  27791  2sqreuopltb  27792  2sqreuopnnlt  27793  2sqreuopnnltb  27794  axlowdimlem6  29525  upgrwlkcompim  30223  clwlkcompbp  30369  mdsldmd1i  32933  fldext2chn  34360  constrextdg2lem  34380  divcnvlin  36498  ditgeq123i  36998  cbvditgvw2  37038  relowlpssretop  38288  2ap1caineq  43196  fsumlessf  46589  climlimsupcex  46779  liminfltlimsupex  46791  liminflelimsupcex  46807  sge0xaddlem2  47444  eubrdm  48106  isgrlim2  49081  iscmgmALT  49321  iscsgrpALT  49323