Theorem breq12i 4102

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 4098	. 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 1398   class class class wbr 4093

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094

This theorem is referenced by:  3brtr3g  4126  3brtr4g  4127  caovord2  6205  ltneg  8701  leneg  8704  inelr  8823  lt2sqi  10952  le2sqi  10953