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Theorem impidc 789
Description: An importation inference for a decidable consequent. (Contributed by Jim Kingdon, 30-Apr-2018.)
Hypothesis
Ref Expression
impidc.1  |-  (DECID  ch  ->  (
ph  ->  ( ps  ->  ch ) ) )
Assertion
Ref Expression
impidc  |-  (DECID  ch  ->  ( -.  ( ph  ->  -. 
ps )  ->  ch ) )

Proof of Theorem impidc
StepHypRef Expression
1 impidc.1 . . . . . 6  |-  (DECID  ch  ->  (
ph  ->  ( ps  ->  ch ) ) )
21imp 122 . . . . 5  |-  ( (DECID  ch 
/\  ph )  ->  ( ps  ->  ch ) )
32con3d 594 . . . 4  |-  ( (DECID  ch 
/\  ph )  ->  ( -.  ch  ->  -.  ps )
)
43ex 113 . . 3  |-  (DECID  ch  ->  (
ph  ->  ( -.  ch  ->  -.  ps ) ) )
54com23 77 . 2  |-  (DECID  ch  ->  ( -.  ch  ->  ( ph  ->  -.  ps )
) )
6 con1dc 787 . 2  |-  (DECID  ch  ->  ( ( -.  ch  ->  (
ph  ->  -.  ps )
)  ->  ( -.  ( ph  ->  -.  ps )  ->  ch ) ) )
75, 6mpd 13 1  |-  (DECID  ch  ->  ( -.  ( ph  ->  -. 
ps )  ->  ch ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102  DECID wdc 776
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663
This theorem depends on definitions:  df-bi 115  df-dc 777
This theorem is referenced by:  simprimdc  790
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