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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-currypeirce | Structured version Visualization version GIF version |
Description: Curry's axiom curryax 892 (a non-intuitionistic positive statement sometimes called a paradox of material implication) implies Peirce's axiom peirce 202 over minimal implicational calculus and the axiomatic definition of disjunction (actually, only the elimination axiom jao 961 via its inference form jaoi 856; the introduction axioms olc 867 and orc 866 are not needed). Note that this theorem shows that actually, the standard instance of curryax 892 implies the standard instance of peirce 202, which is not the case for the converse bj-peircecurry 36473. (Contributed by BJ, 15-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bj-currypeirce | ⊢ ((𝜑 ∨ (𝜑 → 𝜓)) → (((𝜑 → 𝜓) → 𝜑) → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1 6 | . 2 ⊢ (𝜑 → (((𝜑 → 𝜓) → 𝜑) → 𝜑)) | |
2 | pm2.27 42 | . 2 ⊢ ((𝜑 → 𝜓) → (((𝜑 → 𝜓) → 𝜑) → 𝜑)) | |
3 | 1, 2 | jaoi 856 | 1 ⊢ ((𝜑 ∨ (𝜑 → 𝜓)) → (((𝜑 → 𝜓) → 𝜑) → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 846 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 207 df-or 847 |
This theorem is referenced by: (None) |
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