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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 744, condc 749, or notnotrdc 751 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 743 | . 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑)) | |
2 | ax-1 5 | . . 3 ⊢ (𝜑 → (((𝜑 → 𝜓) → 𝜑) → 𝜑)) | |
3 | pm2.21 547 | . . . . 5 ⊢ (¬ 𝜑 → (𝜑 → 𝜓)) | |
4 | 3 | imim1i 54 | . . . 4 ⊢ (((𝜑 → 𝜓) → 𝜑) → (¬ 𝜑 → 𝜑)) |
5 | 4 | com12 27 | . . 3 ⊢ (¬ 𝜑 → (((𝜑 → 𝜓) → 𝜑) → 𝜑)) |
6 | 2, 5 | jaoi 636 | . 2 ⊢ ((𝜑 ∨ ¬ 𝜑) → (((𝜑 → 𝜓) → 𝜑) → 𝜑)) |
7 | 1, 6 | sylbi 114 | 1 ⊢ (DECID 𝜑 → (((𝜑 → 𝜓) → 𝜑) → 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 629 DECID wdc 742 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in2 545 ax-io 630 |
This theorem depends on definitions: df-bi 110 df-dc 743 |
This theorem is referenced by: looinvdc 821 exmoeudc 1963 |
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