Theorem bj-currypeirce 35433

Description: Curry's axiom curryax 893 (a non-intuitionistic positive statement sometimes called a paradox of material implication) implies Peirce's axiom peirce 201 over minimal implicational calculus and the axiomatic definition of disjunction (actually, only the elimination axiom jao 960 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 893 implies the standard instance of peirce 201, which is not the case for the converse bj-peircecurry 35434. (Contributed by BJ, 15-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
bj-currypeirce	⊢ ((𝜑 ∨ (𝜑 → 𝜓)) → (((𝜑 → 𝜓) → 𝜑) → 𝜑))

Proof of Theorem bj-currypeirce

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))

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 206  df-or 847

This theorem is referenced by: (None)