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Theorem pm2.18dc 788
Description: Proof by contradiction for a decidable proposition. Based on Theorem *2.18 of [WhiteheadRussell] p. 103 (also called the Law of Clavius). Intuitionistically it requires a decidability assumption, but compare with pm2.01 581 which does not. (Contributed by Jim Kingdon, 24-Mar-2018.)
Assertion
Ref Expression
pm2.18dc  |-  (DECID  ph  ->  ( ( -.  ph  ->  ph )  ->  ph ) )

Proof of Theorem pm2.18dc
StepHypRef Expression
1 pm2.21 582 . . . 4  |-  ( -. 
ph  ->  ( ph  ->  -.  ( -.  ph  ->  ph ) ) )
21a2i 11 . . 3  |-  ( ( -.  ph  ->  ph )  ->  ( -.  ph  ->  -.  ( -.  ph  ->  ph ) ) )
3 condc 787 . . 3  |-  (DECID  ph  ->  ( ( -.  ph  ->  -.  ( -.  ph  ->  ph ) )  ->  (
( -.  ph  ->  ph )  ->  ph ) ) )
42, 3syl5 32 . 2  |-  (DECID  ph  ->  ( ( -.  ph  ->  ph )  ->  ( ( -.  ph  ->  ph )  ->  ph ) ) )
54pm2.43d 49 1  |-  (DECID  ph  ->  ( ( -.  ph  ->  ph )  ->  ph ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4  DECID wdc 780
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-in2 580  ax-io 665
This theorem depends on definitions:  df-bi 115  df-dc 781
This theorem is referenced by:  pm4.81dc  852
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