Theorem bj-currypeirce 37038

Description: Curry's axiom curryax 906 (a non-intuitionistic positive statement sometimes called a paradox of material implication) implies Peirce's axiom peirce 205 over minimal implicational calculus and the axiomatic definition of disjunction (actually, only the elimination axiom jao 975 via its inference form jaoi 870; the introduction axioms olc 881 and orc 880 are not needed). Note that this theorem shows that actually, the standard instance of curryax 906 implies the standard instance of peirce 205, which is not the case for the converse bj-peircecurry 37039. (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 43	. 2 ⊢ ((𝜑 → 𝜓) → (((𝜑 → 𝜓) → 𝜑) → 𝜑))
3	1, 2	jaoi 870	1 ⊢ ((𝜑 ∨ (𝜑 → 𝜓)) → (((𝜑 → 𝜓) → 𝜑) → 𝜑))

Syntax hints:   → wi 4   ∨ wo 860

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 210  df-or 861

This theorem is referenced by: (None)