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Theorem weeq12d 29533
Description: Equality deduction for well-orders. (Contributed by Stefan O'Rear, 19-Jan-2015.)
Hypotheses
Ref Expression
weeq12d.l  |-  ( ph  ->  R  =  S )
weeq12d.r  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
weeq12d  |-  ( ph  ->  ( R  We  A  <->  S  We  B ) )

Proof of Theorem weeq12d
StepHypRef Expression
1 weeq12d.l . . 3  |-  ( ph  ->  R  =  S )
2 weeq1 4809 . . 3  |-  ( R  =  S  ->  ( R  We  A  <->  S  We  A ) )
31, 2syl 16 . 2  |-  ( ph  ->  ( R  We  A  <->  S  We  A ) )
4 weeq12d.r . . 3  |-  ( ph  ->  A  =  B )
5 weeq2 4810 . . 3  |-  ( A  =  B  ->  ( S  We  A  <->  S  We  B ) )
64, 5syl 16 . 2  |-  ( ph  ->  ( S  We  A  <->  S  We  B ) )
73, 6bitrd 253 1  |-  ( ph  ->  ( R  We  A  <->  S  We  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1370    We wwe 4779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-rex 2801  df-in 3436  df-ss 3443  df-br 4394  df-po 4742  df-so 4743  df-fr 4780  df-we 4782
This theorem is referenced by:  fnwe2lem1  29544  aomclem1  29548  aomclem4  29551  aomclem5  29552  aomclem6  29553
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