Theorem breq12i 4007

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 4003	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷))
4	1, 2, 3	mp2an 426	1 ⊢ (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷)

Syntax hints:   ↔ wb 105   = wceq 1353   class class class wbr 3998

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737  df-un 3131  df-sn 3595  df-pr 3596  df-op 3598  df-br 3999

This theorem is referenced by:  3brtr3g  4031  3brtr4g  4032  caovord2  6037  ltneg  8393  leneg  8396  inelr  8515  lt2sqi  10575  le2sqi  10576