Theorem bj-peircecurry 37427

Description: Peirce's axiom peirce 205 implies Curry's axiom curryax 907 over minimal implicational calculus and the axiomatic definition of disjunction (actually, only the introduction axioms olc 882 and orc 881; the elimination axiom jao 975 is not needed). See bj-currypeirce 37426 for the converse. (Contributed by BJ, 15-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem bj-peircecurry

Step	Hyp	Ref	Expression
1		orc 881	. 2 ⊢ (𝜑 → (𝜑 ∨ (𝜑 → 𝜓)))
2		olc 882	. . 3 ⊢ ((𝜑 → 𝜓) → (𝜑 ∨ (𝜑 → 𝜓)))
3		peirce 205	. . . 4 ⊢ ((((𝜑 ∨ (𝜑 → 𝜓)) → 𝜑) → (𝜑 ∨ (𝜑 → 𝜓))) → (𝜑 ∨ (𝜑 → 𝜓)))
4		peirce 205	. . . . 5 ⊢ (((𝜑 → 𝜓) → 𝜑) → 𝜑)
5		peirceroll 86	. . . . 5 ⊢ ((((𝜑 → 𝜓) → 𝜑) → 𝜑) → (((𝜑 → 𝜓) → (𝜑 ∨ (𝜑 → 𝜓))) → (((𝜑 ∨ (𝜑 → 𝜓)) → 𝜑) → 𝜑)))
6	4, 5	ax-mp 5	. . . 4 ⊢ (((𝜑 → 𝜓) → (𝜑 ∨ (𝜑 → 𝜓))) → (((𝜑 ∨ (𝜑 → 𝜓)) → 𝜑) → 𝜑))
7		peirceroll 86	. . . 4 ⊢ (((((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 69	. . 3 ⊢ (((𝜑 → 𝜓) → (𝜑 ∨ (𝜑 → 𝜓))) → ((𝜑 → (𝜑 ∨ (𝜑 → 𝜓))) → (𝜑 ∨ (𝜑 → 𝜓))))
9	2, 8	ax-mp 5	. 2 ⊢ ((𝜑 → (𝜑 ∨ (𝜑 → 𝜓))) → (𝜑 ∨ (𝜑 → 𝜓)))
10	1, 9	ax-mp 5	1 ⊢ (𝜑 ∨ (𝜑 → 𝜓))

This proof depends on syntax axioms:   → wi 4   ∨ wo 861

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-or 862

This theorem is used by: (None)