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Theorem peircedc 899
Description: Peirce's theorem for a decidable proposition. This odd-looking theorem can be seen as an alternative to exmiddc 821, condc 838, or notnotrdc 828 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 820 . 2 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2 ax-1 6 . . 3 (𝜑 → (((𝜑𝜓) → 𝜑) → 𝜑))
3 pm2.21 606 . . . . 5 𝜑 → (𝜑𝜓))
43imim1i 60 . . . 4 (((𝜑𝜓) → 𝜑) → (¬ 𝜑𝜑))
54com12 30 . . 3 𝜑 → (((𝜑𝜓) → 𝜑) → 𝜑))
62, 5jaoi 705 . 2 ((𝜑 ∨ ¬ 𝜑) → (((𝜑𝜓) → 𝜑) → 𝜑))
71, 6sylbi 120 1 (DECID 𝜑 → (((𝜑𝜓) → 𝜑) → 𝜑))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 697  DECID wdc 819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 604  ax-io 698
This theorem depends on definitions:  df-bi 116  df-dc 820
This theorem is referenced by:  looinvdc  900  exmoeudc  2060
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