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Theorem soeq12d 27103
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 4514 . . 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 4515 . . 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 1652    Or wor 4494
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-in 3319  df-ss 3326  df-br 4205  df-po 4495  df-so 4496
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