Theorem breq12i 5157

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 5153	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷))
4	1, 2, 3	mp2an 692	1 ⊢ (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷)

Syntax hints:   ↔ wb 206   = wceq 1537   class class class wbr 5148

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149

This theorem is referenced by:  3brtr3g  5181  3brtr4g  5182  caovord2  7645  domunfican  9359  ltsonq  11007  ltanq  11009  ltmnq  11010  prlem934  11071  prlem936  11085  ltsosr  11132  ltasr  11138  ltneg  11761  leneg  11764  inelr  12254  lt2sqi  14225  le2sqi  14226  nn0le2msqi  14303  2sqreuop  27521  2sqreuopnn  27522  2sqreuoplt  27523  2sqreuopltb  27524  2sqreuopnnlt  27525  2sqreuopnnltb  27526  axlowdimlem6  28977  upgrwlkcompim  29676  clwlkcompbp  29815  mdsldmd1i  32360  divcnvlin  35713  ditgeq123i  36191  cbvditgvw2  36232  relowlpssretop  37347  2ap1caineq  42127  fsumlessf  45533  climlimsupcex  45725  liminfltlimsupex  45737  liminflelimsupcex  45753  sge0xaddlem2  46390  eubrdm  46986  isgrlim2  47886  iscmgmALT  48068  iscsgrpALT  48070