Theorem merlem11 1681

Description: Step 20 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
merlem11	⊢ ((𝜑 → (𝜑 → 𝜓)) → (𝜑 → 𝜓))

Proof of Theorem merlem11

Step	Hyp	Ref	Expression
1		meredith 1670	. 2 ⊢ (((((𝜑 → 𝜑) → (¬ 𝜑 → ¬ 𝜑)) → 𝜑) → 𝜑) → ((𝜑 → 𝜑) → (𝜑 → 𝜑)))
2		merlem10 1680	. . 3 ⊢ ((𝜑 → (𝜑 → 𝜓)) → ((𝜑 → (𝜑 → 𝜓)) → (𝜑 → 𝜓)))
3		merlem10 1680	. . 3 ⊢ (((𝜑 → (𝜑 → 𝜓)) → ((𝜑 → (𝜑 → 𝜓)) → (𝜑 → 𝜓))) → ((((((𝜑 → 𝜑) → (¬ 𝜑 → ¬ 𝜑)) → 𝜑) → 𝜑) → ((𝜑 → 𝜑) → (𝜑 → 𝜑))) → ((𝜑 → (𝜑 → 𝜓)) → (𝜑 → 𝜓))))
4	2, 3	ax-mp 5	. 2 ⊢ ((((((𝜑 → 𝜑) → (¬ 𝜑 → ¬ 𝜑)) → 𝜑) → 𝜑) → ((𝜑 → 𝜑) → (𝜑 → 𝜑))) → ((𝜑 → (𝜑 → 𝜓)) → (𝜑 → 𝜓)))
5	1, 4	ax-mp 5	1 ⊢ ((𝜑 → (𝜑 → 𝜓)) → (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:  merlem12  1682  merlem13  1683  luk-2  1685  luk-3  1686