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Theorem breqan12rd 4313
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 4312 . 2  |-  ( (
ph  /\  ps )  ->  ( A R C  <-> 
B R D ) )
43ancoms 453 1  |-  ( ( ps  /\  ph )  ->  ( A R C  <-> 
B R D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369   class class class wbr 4297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-br 4298
This theorem is referenced by:  f1oweALT  6566  ledivdiv  10226  xltnegi  11191  ramub1lem1  14092  dvferm1  21462  dvferm2  21464  dvivthlem1  21485  ulmdvlem3  21872  lgsquad  22701  areacirclem4  28492  areacirclem5  28493
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