Theorem breq12i 5068

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 5064	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷))
4	1, 2, 3	mp2an 690	1 ⊢ (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷)

Syntax hints:   ↔ wb 208   = wceq 1533   class class class wbr 5059

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3497  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-br 5060

This theorem is referenced by:  3brtr3g  5092  3brtr4g  5093  caovord2  7354  domunfican  8785  ltsonq  10385  ltanq  10387  ltmnq  10388  prlem934  10449  prlem936  10463  ltsosr  10510  ltasr  10516  ltneg  11134  leneg  11137  inelr  11622  lt2sqi  13546  le2sqi  13547  nn0le2msqi  13621  2sqreuop  26032  2sqreuopnn  26033  2sqreuoplt  26034  2sqreuopltb  26035  2sqreuopnnlt  26036  2sqreuopnnltb  26037  axlowdimlem6  26727  upgrwlkcompim  27418  clwlkcompbp  27557  mdsldmd1i  30102  divcnvlin  32959  relowlpssretop  34639  fsumlessf  41850  climlimsupcex  42042  liminfltlimsupex  42054  liminflelimsupcex  42070  sge0xaddlem2  42709  eubrdm  43264  iscmgmALT  44124  iscsgrpALT  44126