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Theorem soeq12d 26805
Description: Equality deduction for total orderings. (Contributed by Stefan O'Rear, 19-Jan-2015.)
Hypotheses
Ref Expression
weeq12d.l  |-  ( ph  ->  R  =  S )
weeq12d.r  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
soeq12d  |-  ( ph  ->  ( R  Or  A  <->  S  Or  B ) )

Proof of Theorem soeq12d
StepHypRef Expression
1 weeq12d.l . . 3  |-  ( ph  ->  R  =  S )
2 soeq1 4465 . . 3  |-  ( R  =  S  ->  ( R  Or  A  <->  S  Or  A ) )
31, 2syl 16 . 2  |-  ( ph  ->  ( R  Or  A  <->  S  Or  A ) )
4 weeq12d.r . . 3  |-  ( ph  ->  A  =  B )
5 soeq2 4466 . . 3  |-  ( A  =  B  ->  ( S  Or  A  <->  S  Or  B ) )
64, 5syl 16 . 2  |-  ( ph  ->  ( S  Or  A  <->  S  Or  B ) )
73, 6bitrd 245 1  |-  ( ph  ->  ( R  Or  A  <->  S  Or  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    Or wor 4445
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ral 2656  df-in 3272  df-ss 3279  df-br 4156  df-po 4446  df-so 4447
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