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Theorem ecase2d 906
Description: Deduction for elimination by cases. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Dec-2012.)
Hypotheses
Ref Expression
ecase2d.1  |-  ( ph  ->  ps )
ecase2d.2  |-  ( ph  ->  -.  ( ps  /\  ch ) )
ecase2d.3  |-  ( ph  ->  -.  ( ps  /\  th ) )
ecase2d.4  |-  ( ph  ->  ( ta  \/  ( ch  \/  th ) ) )
Assertion
Ref Expression
ecase2d  |-  ( ph  ->  ta )

Proof of Theorem ecase2d
StepHypRef Expression
1 idd 21 . 2  |-  ( ph  ->  ( ta  ->  ta ) )
2 ecase2d.1 . . . 4  |-  ( ph  ->  ps )
3 ecase2d.2 . . . . 5  |-  ( ph  ->  -.  ( ps  /\  ch ) )
43pm2.21d 98 . . . 4  |-  ( ph  ->  ( ( ps  /\  ch )  ->  ta )
)
52, 4mpand 656 . . 3  |-  ( ph  ->  ( ch  ->  ta ) )
6 ecase2d.3 . . . . 5  |-  ( ph  ->  -.  ( ps  /\  th ) )
76pm2.21d 98 . . . 4  |-  ( ph  ->  ( ( ps  /\  th )  ->  ta )
)
82, 7mpand 656 . . 3  |-  ( ph  ->  ( th  ->  ta ) )
95, 8jaod 369 . 2  |-  ( ph  ->  ( ( ch  \/  th )  ->  ta )
)
10 ecase2d.4 . 2  |-  ( ph  ->  ( ta  \/  ( ch  \/  th ) ) )
111, 9, 10mpjaod 370 1  |-  ( ph  ->  ta )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    /\ wa 358
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360
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