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 692	1 ⊢ (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷)

Syntax hints:   ↔ wb 206   = wceq 1540   class class class wbr 5102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103

This theorem is referenced by:  3brtr3g  5135  3brtr4g  5136  caovord2  7581  domunfican  9248  ltsonq  10898  ltanq  10900  ltmnq  10901  prlem934  10962  prlem936  10976  ltsosr  11023  ltasr  11029  ltneg  11654  leneg  11657  lt2sqi  14130  le2sqi  14131  nn0le2msqi  14208  2sqreuop  27406  2sqreuopnn  27407  2sqreuoplt  27408  2sqreuopltb  27409  2sqreuopnnlt  27410  2sqreuopnnltb  27411  axlowdimlem6  28927  upgrwlkcompim  29623  clwlkcompbp  29762  mdsldmd1i  32310  fldext2chn  33711  constrextdg2lem  33731  divcnvlin  35713  ditgeq123i  36190  cbvditgvw2  36230  relowlpssretop  37345  2ap1caineq  42126  fsumlessf  45568  climlimsupcex  45760  liminfltlimsupex  45772  liminflelimsupcex  45788  sge0xaddlem2  46425  eubrdm  47030  isgrlim2  47975  iscmgmALT  48205  iscsgrpALT  48207