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Theorem peircedc 858
Description: Peirce's theorem for a decidable proposition. This odd-looking theorem can be seen as an alternative to exmiddc 782, condc 787, or notnotrdc 789 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 781 . 2  |-  (DECID  ph  <->  ( ph  \/  -.  ph ) )
2 ax-1 5 . . 3  |-  ( ph  ->  ( ( ( ph  ->  ps )  ->  ph )  ->  ph ) )
3 pm2.21 582 . . . . 5  |-  ( -. 
ph  ->  ( ph  ->  ps ) )
43imim1i 59 . . . 4  |-  ( ( ( ph  ->  ps )  ->  ph )  ->  ( -.  ph  ->  ph ) )
54com12 30 . . 3  |-  ( -. 
ph  ->  ( ( (
ph  ->  ps )  ->  ph )  ->  ph )
)
62, 5jaoi 671 . 2  |-  ( (
ph  \/  -.  ph )  ->  ( ( ( ph  ->  ps )  ->  ph )  ->  ph ) )
71, 6sylbi 119 1  |-  (DECID  ph  ->  ( ( ( ph  ->  ps )  ->  ph )  ->  ph ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 664  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:  looinvdc  859  exmoeudc  2011
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