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Theorem ecase13d 28649
Description: Deduction for elimination by cases. (Contributed by Jeff Hankins, 18-Aug-2009.)
Hypotheses
Ref Expression
ecase13d.1  |-  ( ph  ->  -.  ch )
ecase13d.2  |-  ( ph  ->  -.  th )
ecase13d.3  |-  ( ph  ->  ( ch  \/  ps  \/  th ) )
Assertion
Ref Expression
ecase13d  |-  ( ph  ->  ps )

Proof of Theorem ecase13d
StepHypRef Expression
1 ecase13d.2 . 2  |-  ( ph  ->  -.  th )
2 ecase13d.1 . . . 4  |-  ( ph  ->  -.  ch )
3 ecase13d.3 . . . . 5  |-  ( ph  ->  ( ch  \/  ps  \/  th ) )
4 3orass 968 . . . . . 6  |-  ( ( ch  \/  ps  \/  th )  <->  ( ch  \/  ( ps  \/  th )
) )
5 df-or 370 . . . . . 6  |-  ( ( ch  \/  ( ps  \/  th ) )  <-> 
( -.  ch  ->  ( ps  \/  th )
) )
64, 5bitri 249 . . . . 5  |-  ( ( ch  \/  ps  \/  th )  <->  ( -.  ch  ->  ( ps  \/  th ) ) )
73, 6sylib 196 . . . 4  |-  ( ph  ->  ( -.  ch  ->  ( ps  \/  th )
) )
82, 7mpd 15 . . 3  |-  ( ph  ->  ( ps  \/  th ) )
9 orcom 387 . . . 4  |-  ( ( ps  \/  th )  <->  ( th  \/  ps )
)
10 df-or 370 . . . 4  |-  ( ( th  \/  ps )  <->  ( -.  th  ->  ps ) )
119, 10bitri 249 . . 3  |-  ( ( ps  \/  th )  <->  ( -.  th  ->  ps ) )
128, 11sylib 196 . 2  |-  ( ph  ->  ( -.  th  ->  ps ) )
131, 12mpd 15 1  |-  ( ph  ->  ps )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    \/ w3o 964
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 185  df-or 370  df-3or 966
This theorem is referenced by:  ivthALT  28671
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