Theorem breqan12d 4146

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 4135	. 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 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   class class class wbr 4130

This proof depends on 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 proof 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 3715  df-pr 3716  df-op 3718  df-br 4131

This theorem is used by:  breqan12rd  4147  sosng  4848  isoresbr  6015  isoid  6016  isores3  6021  isoini2  6025  ofrfval  6311  oviec  6915  enqbreq2  7724  ltresr2  8207  axpre-ltadd  8253  leltadd  8775  xltneg  10240  lt2sq  11052  le2sq  11053  cnreim  11746  sqrtle  11804  sqrtlt  11805  absext  11831  reefiso  15880  reaplog  15972  logltb  15979  birthdaylem3  16095  lgsquadlem3  16210