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Theorem breqan12rd 4039
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 4038 . 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 1623   class class class wbr 4023
This theorem is referenced by:  f1oweALT  5851  ledivdiv  9645  xltnegi  10543  ramub1lem1  13073  dvferm1  19332  dvferm2  19334  dvivthlem1  19355  ulmdvlem3  19779  lgsquad  20596  areacirclem5  24929  areacirclem6  24930
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024
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