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Theorem peircedc 915
Description: Peirce's theorem for a decidable proposition. This odd-looking theorem can be seen as an alternative to exmiddc 837, condc 854, or notnotrdc 844 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 836 . 2 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2 ax-1 6 . . 3 (𝜑 → (((𝜑𝜓) → 𝜑) → 𝜑))
3 pm2.21 618 . . . . 5 𝜑 → (𝜑𝜓))
43imim1i 60 . . . 4 (((𝜑𝜓) → 𝜑) → (¬ 𝜑𝜑))
54com12 30 . . 3 𝜑 → (((𝜑𝜓) → 𝜑) → 𝜑))
62, 5jaoi 717 . 2 ((𝜑 ∨ ¬ 𝜑) → (((𝜑𝜓) → 𝜑) → 𝜑))
71, 6sylbi 121 1 (DECID 𝜑 → (((𝜑𝜓) → 𝜑) → 𝜑))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 709  DECID wdc 835
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 616  ax-io 710
This theorem depends on definitions:  df-bi 117  df-dc 836
This theorem is referenced by:  looinvdc  916  exmoeudc  2099
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