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Theorem peircedc 820
 Description: Peirce's theorem for a decidable proposition. This odd-looking theorem can be seen as an alternative to exmiddc 744, condc 749, or notnotrdc 751 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 743 . 2 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2 ax-1 5 . . 3 (𝜑 → (((𝜑𝜓) → 𝜑) → 𝜑))
3 pm2.21 547 . . . . 5 𝜑 → (𝜑𝜓))
43imim1i 54 . . . 4 (((𝜑𝜓) → 𝜑) → (¬ 𝜑𝜑))
54com12 27 . . 3 𝜑 → (((𝜑𝜓) → 𝜑) → 𝜑))
62, 5jaoi 636 . 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 629  DECID wdc 742 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 545  ax-io 630 This theorem depends on definitions:  df-bi 110  df-dc 743 This theorem is referenced by:  looinvdc  821  exmoeudc  1963
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