![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 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.) |
Ref | Expression |
---|---|
peircedc | ⊢ (DECID 𝜑 → (((𝜑 → 𝜓) → 𝜑) → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dc 835 | . 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑)) | |
2 | ax-1 6 | . . 3 ⊢ (𝜑 → (((𝜑 → 𝜓) → 𝜑) → 𝜑)) | |
3 | pm2.21 617 | . . . . 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 716 | . 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 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 |
Copyright terms: Public domain | W3C validator |