Theorem bj-currypeirce 36880

Description: Curry's axiom curryax 900 (a non-intuitionistic positive statement sometimes called a paradox of material implication) implies Peirce's axiom peirce 204 over minimal implicational calculus and the axiomatic definition of disjunction (actually, only the elimination axiom jao 969 via its inference form jaoi 864; the introduction axioms olc 875 and orc 874 are not needed). Note that this theorem shows that actually, the standard instance of curryax 900 implies the standard instance of peirce 204, which is not the case for the converse bj-peircecurry 36881. (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 864	1 ⊢ ((𝜑 ∨ (𝜑 → 𝜓)) → (((𝜑 → 𝜓) → 𝜑) → 𝜑))

Syntax hints:   → wi 4   ∨ wo 854

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 209  df-or 855

This theorem is referenced by: (None)