Theorem bj-peircecurry

Description: Peirce's axiom peirce implies Curry's axiom curryax over minimal implicational calculus and the axiomatic definition of disjunction (actually, only the introduction axioms olc and orc ; the elimination axiom jao is not needed). See bj-currypeirce for the converse. (Contributed by BJ, 15-Jun-2021) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		orc	⊢ ( 𝜑 → ( 𝜑 ∨ ( 𝜑 → 𝜓 ) ) )
2		olc	⊢ ( ( 𝜑 → 𝜓 ) → ( 𝜑 ∨ ( 𝜑 → 𝜓 ) ) )
3		peirce	⊢ ( ( ( ( 𝜑 ∨ ( 𝜑 → 𝜓 ) ) → 𝜑 ) → ( 𝜑 ∨ ( 𝜑 → 𝜓 ) ) ) → ( 𝜑 ∨ ( 𝜑 → 𝜓 ) ) )
4		peirce	⊢ ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜑 )
5		peirceroll	⊢ ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 ∨ ( 𝜑 → 𝜓 ) ) ) → ( ( ( 𝜑 ∨ ( 𝜑 → 𝜓 ) ) → 𝜑 ) → 𝜑 ) ) )
6	4 5	ax-mp	⊢ ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 ∨ ( 𝜑 → 𝜓 ) ) ) → ( ( ( 𝜑 ∨ ( 𝜑 → 𝜓 ) ) → 𝜑 ) → 𝜑 ) )
7		peirceroll	⊢ ( ( ( ( ( 𝜑 ∨ ( 𝜑 → 𝜓 ) ) → 𝜑 ) → ( 𝜑 ∨ ( 𝜑 → 𝜓 ) ) ) → ( 𝜑 ∨ ( 𝜑 → 𝜓 ) ) ) → ( ( ( ( 𝜑 ∨ ( 𝜑 → 𝜓 ) ) → 𝜑 ) → 𝜑 ) → ( ( 𝜑 → ( 𝜑 ∨ ( 𝜑 → 𝜓 ) ) ) → ( 𝜑 ∨ ( 𝜑 → 𝜓 ) ) ) ) )
8	3 6 7	mpsyl	⊢ ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 ∨ ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → ( 𝜑 ∨ ( 𝜑 → 𝜓 ) ) ) → ( 𝜑 ∨ ( 𝜑 → 𝜓 ) ) ) )
9	2 8	ax-mp	⊢ ( ( 𝜑 → ( 𝜑 ∨ ( 𝜑 → 𝜓 ) ) ) → ( 𝜑 ∨ ( 𝜑 → 𝜓 ) ) )
10	1 9	ax-mp	⊢ ( 𝜑 ∨ ( 𝜑 → 𝜓 ) )