Theorem bj-currypeirce 36552

Description: Curry's axiom curryax 894 (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 963 via its inference form jaoi 858; the introduction axioms olc 869 and orc 868 are not needed). Note that this theorem shows that actually, the standard instance of curryax 894 implies the standard instance of peirce 202, which is not the case for the converse bj-peircecurry 36553. (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 858	1 ⊢ ((𝜑 ∨ (𝜑 → 𝜓)) → (((𝜑 → 𝜓) → 𝜑) → 𝜑))

Syntax hints:   → wi 4   ∨ wo 848

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 849

This theorem is referenced by: (None)