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Theorem ex-natded5.13-2 20823
Description: A more efficient proof of Theorem 5.13 of [Clemente] p. 20. Compare with ex-natded5.13 20822. (Contributed by Mario Carneiro, 9-Feb-2017.)
Hypotheses
Ref Expression
ex-natded5.13.1  |-  ( ph  ->  ( ps  \/  ch ) )
ex-natded5.13.2  |-  ( ph  ->  ( ps  ->  th )
)
ex-natded5.13.3  |-  ( ph  ->  ( -.  ta  ->  -. 
ch ) )
Assertion
Ref Expression
ex-natded5.13-2  |-  ( ph  ->  ( th  \/  ta ) )

Proof of Theorem ex-natded5.13-2
StepHypRef Expression
1 ex-natded5.13.1 . 2  |-  ( ph  ->  ( ps  \/  ch ) )
2 ex-natded5.13.2 . . 3  |-  ( ph  ->  ( ps  ->  th )
)
3 ex-natded5.13.3 . . . 4  |-  ( ph  ->  ( -.  ta  ->  -. 
ch ) )
43con4d 99 . . 3  |-  ( ph  ->  ( ch  ->  ta ) )
52, 4orim12d 813 . 2  |-  ( ph  ->  ( ( ps  \/  ch )  ->  ( th  \/  ta ) ) )
61, 5mpd 16 1  |-  ( ph  ->  ( th  \/  ta ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    \/ wo 359
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362
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