Theorem breq12i 5175

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 5171	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷))
4	1, 2, 3	mp2an 691	1 ⊢ (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷)

Syntax hints:   ↔ wb 206   = wceq 1537   class class class wbr 5166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167

This theorem is referenced by:  3brtr3g  5199  3brtr4g  5200  caovord2  7662  domunfican  9389  ltsonq  11038  ltanq  11040  ltmnq  11041  prlem934  11102  prlem936  11116  ltsosr  11163  ltasr  11169  ltneg  11790  leneg  11793  inelr  12283  lt2sqi  14238  le2sqi  14239  nn0le2msqi  14316  2sqreuop  27524  2sqreuopnn  27525  2sqreuoplt  27526  2sqreuopltb  27527  2sqreuopnnlt  27528  2sqreuopnnltb  27529  axlowdimlem6  28980  upgrwlkcompim  29679  clwlkcompbp  29818  mdsldmd1i  32363  divcnvlin  35695  ditgeq123i  36173  cbvditgvw2  36215  relowlpssretop  37330  2ap1caineq  42102  fsumlessf  45498  climlimsupcex  45690  liminfltlimsupex  45702  liminflelimsupcex  45718  sge0xaddlem2  46355  eubrdm  46951  isgrlim2  47807  iscmgmALT  47947  iscsgrpALT  47949