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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 831, condc 848, or notnotrdc 838 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 830 | . 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑)) | |
2 | ax-1 6 | . . 3 ⊢ (𝜑 → (((𝜑 → 𝜓) → 𝜑) → 𝜑)) | |
3 | pm2.21 612 | . . . . 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 711 | . 2 ⊢ ((𝜑 ∨ ¬ 𝜑) → (((𝜑 → 𝜓) → 𝜑) → 𝜑)) |
7 | 1, 6 | sylbi 120 | 1 ⊢ (DECID 𝜑 → (((𝜑 → 𝜓) → 𝜑) → 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 703 DECID wdc 829 |
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 610 ax-io 704 |
This theorem depends on definitions: df-bi 116 df-dc 830 |
This theorem is referenced by: looinvdc 910 exmoeudc 2082 |
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