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Theorem peircedc 882
Description: Peirce's theorem for a decidable proposition. This odd-looking theorem can be seen as an alternative to exmiddc 804, condc 821, or notnotrdc 811 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 803 . 2  |-  (DECID  ph  <->  ( ph  \/  -.  ph ) )
2 ax-1 6 . . 3  |-  ( ph  ->  ( ( ( ph  ->  ps )  ->  ph )  ->  ph ) )
3 pm2.21 589 . . . . 5  |-  ( -. 
ph  ->  ( ph  ->  ps ) )
43imim1i 60 . . . 4  |-  ( ( ( ph  ->  ps )  ->  ph )  ->  ( -.  ph  ->  ph ) )
54com12 30 . . 3  |-  ( -. 
ph  ->  ( ( (
ph  ->  ps )  ->  ph )  ->  ph )
)
62, 5jaoi 688 . 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 680  DECID wdc 802
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 587  ax-io 681
This theorem depends on definitions:  df-bi 116  df-dc 803
This theorem is referenced by:  looinvdc  883  exmoeudc  2038
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