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Theorem peircedc 909
Description: Peirce's theorem for a decidable proposition. This odd-looking theorem can be seen as an alternative to exmiddc 831, condc 848, or notnotrdc 838 in the sense of expressing the "difference" between an intuitionistic system of propositional calculus and a classical system. In intuitionistic logic, it only holds for decidable propositions. (Contributed by Jim Kingdon, 3-Jul-2018.)
Assertion
Ref Expression
peircedc  |-  (DECID  ph  ->  ( ( ( ph  ->  ps )  ->  ph )  ->  ph ) )

Proof of Theorem peircedc
StepHypRef Expression
1 df-dc 830 . 2  |-  (DECID  ph  <->  ( ph  \/  -.  ph ) )
2 ax-1 6 . . 3  |-  ( ph  ->  ( ( ( ph  ->  ps )  ->  ph )  ->  ph ) )
3 pm2.21 612 . . . . 5  |-  ( -. 
ph  ->  ( ph  ->  ps ) )
43imim1i 60 . . . 4  |-  ( ( ( ph  ->  ps )  ->  ph )  ->  ( -.  ph  ->  ph ) )
54com12 30 . . 3  |-  ( -. 
ph  ->  ( ( (
ph  ->  ps )  ->  ph )  ->  ph )
)
62, 5jaoi 711 . 2  |-  ( (
ph  \/  -.  ph )  ->  ( ( ( ph  ->  ps )  ->  ph )  ->  ph ) )
71, 6sylbi 120 1  |-  (DECID  ph  ->  ( ( ( ph  ->  ps )  ->  ph )  ->  ph ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 703  DECID wdc 829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 610  ax-io 704
This theorem depends on definitions:  df-bi 116  df-dc 830
This theorem is referenced by:  looinvdc  910  exmoeudc  2082
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