Theorem merlem4 1674

Description: Step 8 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
merlem4	⊢ (𝜏 → ((𝜏 → 𝜑) → (𝜃 → 𝜑)))

Proof of Theorem merlem4

Step	Hyp	Ref	Expression
1		meredith 1670	. 2 ⊢ (((((𝜑 → 𝜑) → (¬ 𝜃 → ¬ 𝜃)) → 𝜃) → 𝜏) → ((𝜏 → 𝜑) → (𝜃 → 𝜑)))
2		merlem3 1673	. 2 ⊢ ((((((𝜑 → 𝜑) → (¬ 𝜃 → ¬ 𝜃)) → 𝜃) → 𝜏) → ((𝜏 → 𝜑) → (𝜃 → 𝜑))) → (𝜏 → ((𝜏 → 𝜑) → (𝜃 → 𝜑))))
3	1, 2	ax-mp 5	1 ⊢ (𝜏 → ((𝜏 → 𝜑) → (𝜃 → 𝜑)))

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:  merlem5  1675  merlem6  1676  merlem7  1677  merlem12  1682  luk-2  1685