Theorem breqan12d 3866

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 3856	. 2 |- ( ( A = B /\ C = D ) -> ( A R C <-> B R D ) )
4	1, 2, 3	syl2an 284	1 |- ( ( ph /\ ps ) -> ( A R C <-> B R D ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1290   class class class wbr 3851

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-un 3004  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852

This theorem is referenced by:  breqan12rd  3867  sosng  4524  isoresbr  5602  isoid  5603  isores3  5608  isoini2  5612  ofrfval  5878  oviec  6412  enqbreq2  6977  ltresr2  7438  axpre-ltadd  7482  leltadd  7986  xltneg  9359  lt2sq  10089  le2sq  10090  sqrtle  10530  sqrtlt  10531  absext  10557