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Theorem breqan12rd 3916
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 3915 . 2  |-  ( (
ph  /\  ps )  ->  ( A R C  <-> 
B R D ) )
43ancoms 266 1  |-  ( ( ps  /\  ph )  ->  ( A R C  <-> 
B R D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1316   class class class wbr 3899
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900
This theorem is referenced by:  xltnegi  9586
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