Theorem breqan12d 4099

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 4088	. 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 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:  breqan12rd  4100  sosng  4792  isoresbr  5939  isoid  5940  isores3  5945  isoini2  5949  ofrfval  6233  oviec  6796  enqbreq2  7552  ltresr2  8035  axpre-ltadd  8081  leltadd  8602  xltneg  10040  lt2sq  10843  le2sq  10844  cnreim  11497  sqrtle  11555  sqrtlt  11556  absext  11582  reefiso  15459  logltb  15556  lgsquadlem3  15766