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Theorem jaddc 802
Description: Deduction forming an implication from the antecedents of two premises, where a decidable antecedent is negated. (Contributed by Jim Kingdon, 26-Mar-2018.)
Hypotheses
Ref Expression
jaddc.1  |-  ( ph  ->  (DECID  ps  ->  ( -.  ps  ->  th ) ) )
jaddc.2  |-  ( ph  ->  ( ch  ->  th )
)
Assertion
Ref Expression
jaddc  |-  ( ph  ->  (DECID  ps  ->  ( ( ps  ->  ch )  ->  th ) ) )

Proof of Theorem jaddc
StepHypRef Expression
1 jaddc.2 . . 3  |-  ( ph  ->  ( ch  ->  th )
)
21imim2d 54 . 2  |-  ( ph  ->  ( ( ps  ->  ch )  ->  ( ps  ->  th ) ) )
3 jaddc.1 . . 3  |-  ( ph  ->  (DECID  ps  ->  ( -.  ps  ->  th ) ) )
4 pm2.6dc 800 . . 3  |-  (DECID  ps  ->  ( ( -.  ps  ->  th )  ->  ( ( ps  ->  th )  ->  th )
) )
53, 4sylcom 28 . 2  |-  ( ph  ->  (DECID  ps  ->  ( ( ps  ->  th )  ->  th )
) )
62, 5syl5d 68 1  |-  ( ph  ->  (DECID  ps  ->  ( ( ps  ->  ch )  ->  th ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4  DECID wdc 783
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668
This theorem depends on definitions:  df-bi 116  df-dc 784
This theorem is referenced by:  pm2.54dc  831
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