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Theorem breqan12d 4034
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
StepHypRef Expression
1 breq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 breqan12i.2 . 2  |-  ( ps 
->  C  =  D
)
3 breq12 4023 . 2  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A R C  <-> 
B R D ) )
41, 2, 3syl2an 289 1  |-  ( (
ph  /\  ps )  ->  ( A R C  <-> 
B R D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1364   class class class wbr 4018
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019
This theorem is referenced by:  breqan12rd  4035  sosng  4717  isoresbr  5831  isoid  5832  isores3  5837  isoini2  5841  ofrfval  6115  oviec  6667  enqbreq2  7386  ltresr2  7869  axpre-ltadd  7915  leltadd  8434  xltneg  9866  lt2sq  10625  le2sq  10626  cnreim  11019  sqrtle  11077  sqrtlt  11078  absext  11104  reefiso  14658  logltb  14755
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