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Theorem breqan12rd 4141
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 4140 . 2  |-  ( (
ph  /\  ps )  ->  ( A R C  <-> 
B R D ) )
43ancoms 439 1  |-  ( ( ps  /\  ph )  ->  ( A R C  <-> 
B R D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1647   class class class wbr 4125
This theorem is referenced by:  f1oweALT  5974  ledivdiv  9792  xltnegi  10695  ramub1lem1  13281  dvferm1  19547  dvferm2  19549  dvivthlem1  19570  ulmdvlem3  19996  lgsquad  20819  areacirclem5  25789  areacirclem6  25790
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-br 4126
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