Theorem merlem10 1684

Description: Step 19 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
merlem10	⊢ ((𝜑 → (𝜑 → 𝜓)) → (𝜃 → (𝜑 → 𝜓)))

Proof of Theorem merlem10

Step	Hyp	Ref	Expression
1		meredith 1674	. 2 ⊢ (((((𝜑 → 𝜑) → (¬ 𝜑 → ¬ 𝜑)) → 𝜑) → 𝜑) → ((𝜑 → 𝜑) → (𝜑 → 𝜑)))
2		meredith 1674	. . 3 ⊢ ((((((𝜑 → 𝜓) → 𝜑) → (¬ 𝜑 → ¬ 𝜃)) → 𝜑) → 𝜑) → ((𝜑 → (𝜑 → 𝜓)) → (𝜃 → (𝜑 → 𝜓))))
3		merlem9 1683	. . 3 ⊢ (((((((𝜑 → 𝜓) → 𝜑) → (¬ 𝜑 → ¬ 𝜃)) → 𝜑) → 𝜑) → ((𝜑 → (𝜑 → 𝜓)) → (𝜃 → (𝜑 → 𝜓)))) → ((((((𝜑 → 𝜑) → (¬ 𝜑 → ¬ 𝜑)) → 𝜑) → 𝜑) → ((𝜑 → 𝜑) → (𝜑 → 𝜑))) → ((𝜑 → (𝜑 → 𝜓)) → (𝜃 → (𝜑 → 𝜓)))))
4	2, 3	ax-mp 5	. 2 ⊢ ((((((𝜑 → 𝜑) → (¬ 𝜑 → ¬ 𝜑)) → 𝜑) → 𝜑) → ((𝜑 → 𝜑) → (𝜑 → 𝜑))) → ((𝜑 → (𝜑 → 𝜓)) → (𝜃 → (𝜑 → 𝜓))))
5	1, 4	ax-mp 5	1 ⊢ ((𝜑 → (𝜑 → 𝜓)) → (𝜃 → (𝜑 → 𝜓)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4

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

This theorem is used by:  merlem11  1685