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Theorem weeq12d 31224
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 4856 . . 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 4857 . . 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 1398    We wwe 4826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-ral 2809  df-rex 2810  df-in 3468  df-ss 3475  df-br 4440  df-po 4789  df-so 4790  df-fr 4827  df-we 4829
This theorem is referenced by:  fnwe2lem1  31235  aomclem1  31239  aomclem4  31242  aomclem5  31243  aomclem6  31244
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