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Theorem contrd 29068
Description: A proof by contradiction, in deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
Hypotheses
Ref Expression
contrd.1  |-  ( ph  ->  ( -.  ps  ->  ch ) )
contrd.2  |-  ( ph  ->  ( -.  ps  ->  -. 
ch ) )
Assertion
Ref Expression
contrd  |-  ( ph  ->  ps )

Proof of Theorem contrd
StepHypRef Expression
1 contrd.1 . . 3  |-  ( ph  ->  ( -.  ps  ->  ch ) )
2 contrd.2 . . 3  |-  ( ph  ->  ( -.  ps  ->  -. 
ch ) )
31, 2jcad 533 . 2  |-  ( ph  ->  ( -.  ps  ->  ( ch  /\  -.  ch ) ) )
4 pm2.24 109 . . . . 5  |-  ( ch 
->  ( -.  ch  ->  ps ) )
54imp 429 . . . 4  |-  ( ( ch  /\  -.  ch )  ->  ps )
65imim2i 14 . . 3  |-  ( ( -.  ps  ->  ( ch  /\  -.  ch )
)  ->  ( -.  ps  ->  ps ) )
7 pm2.18 110 . . 3  |-  ( ( -.  ps  ->  ps )  ->  ps )
86, 7syl 16 . 2  |-  ( ( -.  ps  ->  ( ch  /\  -.  ch )
)  ->  ps )
93, 8syl 16 1  |-  ( ph  ->  ps )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369
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-an 371
This theorem is referenced by:  mpt2bi123f  29143  mptbi12f  29147  ac6s6  29152
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