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Theorem breqan12rd 3861
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
breqan12rd  |-  ( ( ps  /\  ph )  ->  ( A R C  <-> 
B R D ) )

Proof of Theorem breqan12rd
StepHypRef Expression
1 breq1d.1 . . 3  |-  ( ph  ->  A  =  B )
2 breqan12i.2 . . 3  |-  ( ps 
->  C  =  D
)
31, 2breqan12d 3860 . 2  |-  ( (
ph  /\  ps )  ->  ( A R C  <-> 
B R D ) )
43ancoms 264 1  |-  ( ( ps  /\  ph )  ->  ( A R C  <-> 
B R D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289   class class class wbr 3845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846
This theorem is referenced by:  xltnegi  9297
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