Theorem breqan12d 4144

Description: Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)

Hypotheses

Ref	Expression
breq1d.1	|- ( ph -> A = B )
breqan12i.2	|- ( ps -> C = D )

Assertion

Ref	Expression
breqan12d	|- ( ( ph /\ ps ) -> ( A R C <-> B R D ) )

Proof of Theorem breqan12d

Step	Hyp	Ref	Expression
1		breq1d.1	. 2 |- ( ph -> A = B )
2		breqan12i.2	. 2 |- ( ps -> C = D )
3		breq12 4133	. 2 |- ( ( A = B /\ C = D ) -> ( A R C <-> B R D ) )
4	1, 2, 3	syl2an 289	1 |- ( ( ph /\ ps ) -> ( A R C <-> B R D ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   class class class wbr 4128

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3714  df-pr 3715  df-op 3717  df-br 4129

This theorem is referenced by:  breqan12rd  4145  sosng  4846  isoresbr  6009  isoid  6010  isores3  6015  isoini2  6019  ofrfval  6305  oviec  6909  enqbreq2  7718  ltresr2  8201  axpre-ltadd  8247  leltadd  8769  xltneg  10221  lt2sq  11033  le2sq  11034  cnreim  11727  sqrtle  11785  sqrtlt  11786  absext  11812  reefiso  15861  logltb  15958  birthdaylem3  16072  lgsquadlem3  16181