Theorem breq12i 5098

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 5094	. 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 5089

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 2113  ax-9 2121  ax-ext 2703

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 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090

This theorem is referenced by:  3brtr3g  5122  3brtr4g  5123  caovord2  7558  domunfican  9206  ltsonq  10860  ltanq  10862  ltmnq  10863  prlem934  10924  prlem936  10938  ltsosr  10985  ltasr  10991  ltneg  11617  leneg  11620  lt2sqi  14096  le2sqi  14097  nn0le2msqi  14174  2sqreuop  27400  2sqreuopnn  27401  2sqreuoplt  27402  2sqreuopltb  27403  2sqreuopnnlt  27404  2sqreuopnnltb  27405  axlowdimlem6  28925  upgrwlkcompim  29621  clwlkcompbp  29760  mdsldmd1i  32311  fldext2chn  33741  constrextdg2lem  33761  divcnvlin  35777  ditgeq123i  36253  cbvditgvw2  36293  relowlpssretop  37408  2ap1caineq  42237  fsumlessf  45676  climlimsupcex  45866  liminfltlimsupex  45878  liminflelimsupcex  45894  sge0xaddlem2  46531  eubrdm  47135  isgrlim2  48082  iscmgmALT  48323  iscsgrpALT  48325