Theorem merlem12 1652

Description: Step 28 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
merlem12	⊢ (((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → 𝜑) → 𝜑)

Proof of Theorem merlem12

Step	Hyp	Ref	Expression
1		merlem5 1645	. . . 4 ⊢ ((𝜒 → 𝜒) → (¬ ¬ 𝜒 → 𝜒))
2		merlem2 1642	. . . 4 ⊢ (((𝜒 → 𝜒) → (¬ ¬ 𝜒 → 𝜒)) → (𝜃 → (¬ ¬ 𝜒 → 𝜒)))
3	1, 2	ax-mp 5	. . 3 ⊢ (𝜃 → (¬ ¬ 𝜒 → 𝜒))
4		merlem4 1644	. . 3 ⊢ ((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → (((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → 𝜑) → (((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → 𝜑) → 𝜑)))
5	3, 4	ax-mp 5	. 2 ⊢ (((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → 𝜑) → (((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → 𝜑) → 𝜑))
6		merlem11 1651	. 2 ⊢ ((((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → 𝜑) → (((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → 𝜑) → 𝜑)) → (((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → 𝜑) → 𝜑))
7	5, 6	ax-mp 5	1 ⊢ (((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → 𝜑) → 𝜑)

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:  merlem13  1653