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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
StepHypRef 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 ) )
41, 2, 3syl2an 284 1  |-  ( (
ph  /\  ps )  ->  ( A R C  <-> 
B R D ) )
Colors of variables: wff set class
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
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