Theorem bj-currypeirce

Description: Curry's axiom curryax (a non-intuitionistic positive statement sometimes called a paradox of material implication) implies Peirce's axiom peirce over minimal implicational calculus and the axiomatic definition of disjunction (actually, only the elimination axiom jao via its inference form jaoi ; the introduction axioms olc and orc are not needed). Note that this theorem shows that actually, the standard instance of curryax implies the standard instance of peirce , which is not the case for the converse bj-peircecurry . (Contributed by BJ, 15-Jun-2021) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		ax-1	⊢ ( 𝜑 → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜑 ) )
2		pm2.27	⊢ ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜑 ) )
3	1 2	jaoi	⊢ ( ( 𝜑 ∨ ( 𝜑 → 𝜓 ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜑 ) )