Theorem merlem2 1643

Description: Step 4 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
merlem2	⊢ (((𝜑 → 𝜑) → 𝜒) → (𝜃 → 𝜒))

Proof of Theorem merlem2

Step	Hyp	Ref	Expression
1		merlem1 1642	. 2 ⊢ ((((𝜒 → 𝜒) → (¬ 𝜑 → ¬ 𝜃)) → 𝜑) → (𝜑 → 𝜑))
2		meredith 1641	. 2 ⊢ (((((𝜒 → 𝜒) → (¬ 𝜑 → ¬ 𝜃)) → 𝜑) → (𝜑 → 𝜑)) → (((𝜑 → 𝜑) → 𝜒) → (𝜃 → 𝜒)))
3	1, 2	ax-mp 5	1 ⊢ (((𝜑 → 𝜑) → 𝜒) → (𝜃 → 𝜒))

Syntax hints:  ¬ wn 3   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is referenced by:  merlem3  1644  merlem12  1653