Theorem breqan12d 4003

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 3992	. 2 |- ( ( A = B /\ C = D ) -> ( A R C <-> B R D ) )
4	1, 2, 3	syl2an 287	1 |- ( ( ph /\ ps ) -> ( A R C <-> B R D ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   class class class wbr 3987

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988

This theorem is referenced by:  breqan12rd  4004  sosng  4682  isoresbr  5785  isoid  5786  isores3  5791  isoini2  5795  ofrfval  6066  oviec  6615  enqbreq2  7306  ltresr2  7789  axpre-ltadd  7835  leltadd  8353  xltneg  9780  lt2sq  10536  le2sq  10537  cnreim  10929  sqrtle  10987  sqrtlt  10988  absext  11014  reefiso  13413  logltb  13510