Theorem merlem8 1682

Description: Step 15 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
merlem8	⊢ (((𝜓 → 𝜒) → 𝜃) → (((𝜒 → 𝜏) → (¬ 𝜃 → ¬ 𝜓)) → 𝜃))

Proof of Theorem merlem8

Step	Hyp	Ref	Expression
1		meredith 1674	. 2 ⊢ (((((𝜑 → 𝜑) → (¬ 𝜑 → ¬ 𝜑)) → 𝜑) → 𝜑) → ((𝜑 → 𝜑) → (𝜑 → 𝜑)))
2		merlem7 1681	. 2 ⊢ ((((((𝜑 → 𝜑) → (¬ 𝜑 → ¬ 𝜑)) → 𝜑) → 𝜑) → ((𝜑 → 𝜑) → (𝜑 → 𝜑))) → (((𝜓 → 𝜒) → 𝜃) → (((𝜒 → 𝜏) → (¬ 𝜃 → ¬ 𝜓)) → 𝜃)))
3	1, 2	ax-mp 5	1 ⊢ (((𝜓 → 𝜒) → 𝜃) → (((𝜒 → 𝜏) → (¬ 𝜃 → ¬ 𝜓)) → 𝜃))

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:  merlem9  1683