Theorem breq12i 4092

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 4088	. 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 1395   class class class wbr 4083

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084

This theorem is referenced by:  3brtr3g  4116  3brtr4g  4117  caovord2  6178  ltneg  8609  leneg  8612  inelr  8731  lt2sqi  10849  le2sqi  10850