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Theorem peircedc 813
 Description: Peirce's theorem for a decidable proposition. This odd-looking theorem can be seen as an alternative to exmiddc 735, condc 740, or notnotdc 759 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 φ → (((φψ) → φ) → φ))

Proof of Theorem peircedc
StepHypRef Expression
1 df-dc 734 . 2 (DECID φ ↔ (φ ¬ φ))
2 ax-1 5 . . 3 (φ → (((φψ) → φ) → φ))
3 pm2.21 535 . . . . 5 φ → (φψ))
43imim1i 54 . . . 4 (((φψ) → φ) → (¬ φφ))
54com12 27 . . 3 φ → (((φψ) → φ) → φ))
62, 5jaoi 623 . 2 ((φ ¬ φ) → (((φψ) → φ) → φ))
71, 6sylbi 114 1 (DECID φ → (((φψ) → φ) → φ))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 616  DECID wdc 733 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 533  ax-io 617 This theorem depends on definitions:  df-bi 110  df-dc 734 This theorem is referenced by:  looinvdc  814  exmoeudc  1945
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