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Theorem peircedc 914
Description: Peirce's theorem for a decidable proposition. This odd-looking theorem can be seen as an alternative to exmiddc 836, condc 853, or notnotrdc 843 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 835 . 2 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2 ax-1 6 . . 3 (𝜑 → (((𝜑𝜓) → 𝜑) → 𝜑))
3 pm2.21 617 . . . . 5 𝜑 → (𝜑𝜓))
43imim1i 60 . . . 4 (((𝜑𝜓) → 𝜑) → (¬ 𝜑𝜑))
54com12 30 . . 3 𝜑 → (((𝜑𝜓) → 𝜑) → 𝜑))
62, 5jaoi 716 . 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 708  DECID wdc 834
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 615  ax-io 709
This theorem depends on definitions:  df-bi 117  df-dc 835
This theorem is referenced by:  looinvdc  915  exmoeudc  2089
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