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Mirrors > Home > ILE Home > Th. List > peircedc | GIF version |
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.) |
Ref | Expression |
---|---|
peircedc | ⊢ (DECID 𝜑 → (((𝜑 → 𝜓) → 𝜑) → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dc 836 | . 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑)) | |
2 | ax-1 6 | . . 3 ⊢ (𝜑 → (((𝜑 → 𝜓) → 𝜑) → 𝜑)) | |
3 | pm2.21 618 | . . . . 5 ⊢ (¬ 𝜑 → (𝜑 → 𝜓)) | |
4 | 3 | imim1i 60 | . . . 4 ⊢ (((𝜑 → 𝜓) → 𝜑) → (¬ 𝜑 → 𝜑)) |
5 | 4 | com12 30 | . . 3 ⊢ (¬ 𝜑 → (((𝜑 → 𝜓) → 𝜑) → 𝜑)) |
6 | 2, 5 | jaoi 717 | . 2 ⊢ ((𝜑 ∨ ¬ 𝜑) → (((𝜑 → 𝜓) → 𝜑) → 𝜑)) |
7 | 1, 6 | sylbi 121 | 1 ⊢ (DECID 𝜑 → (((𝜑 → 𝜓) → 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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