Theorem bj-currypeirce 34005

Description: Curry's axiom curryax 891 (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 958 via its inference form jaoi 854; the introduction axioms olc 865 and orc 864 are not needed). Note that this theorem shows that actually, the standard instance of curryax 891 implies the standard instance of peirce 205, which is not the case for the converse bj-peircecurry 34006. (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 854	1 ⊢ ((𝜑 ∨ (𝜑 → 𝜓)) → (((𝜑 → 𝜓) → 𝜑) → 𝜑))

Syntax hints:   → wi 4   ∨ wo 844

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 845

This theorem is referenced by: (None)