Theorem breq12i 4068

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	|- ( A R C <-> B R D )

Proof of Theorem breq12i

Step	Hyp	Ref	Expression
1		breq1i.1	. 2 |- A = B
2		breq12i.2	. 2 |- C = D
3		breq12 4064	. 2 |- ( ( A = B /\ C = D ) -> ( A R C <-> B R D ) )
4	1, 2, 3	mp2an 426	1 |- ( A R C <-> B R D )

Syntax hints:   <-> wb 105   = wceq 1373   class class class wbr 4059

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060

This theorem is referenced by:  3brtr3g  4092  3brtr4g  4093  caovord2  6142  ltneg  8570  leneg  8573  inelr  8692  lt2sqi  10809  le2sqi  10810