Theorem merlem3 1643

Description: Step 7 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
merlem3	⊢ (((𝜓 → 𝜒) → 𝜑) → (𝜒 → 𝜑))

Proof of Theorem merlem3

Step	Hyp	Ref	Expression
1		merlem2 1642	. . . 4 ⊢ (((¬ 𝜒 → ¬ 𝜒) → (¬ 𝜒 → ¬ 𝜒)) → ((𝜑 → 𝜑) → (¬ 𝜒 → ¬ 𝜒)))
2		merlem2 1642	. . . 4 ⊢ ((((¬ 𝜒 → ¬ 𝜒) → (¬ 𝜒 → ¬ 𝜒)) → ((𝜑 → 𝜑) → (¬ 𝜒 → ¬ 𝜒))) → ((((𝜒 → 𝜑) → (¬ 𝜓 → ¬ 𝜓)) → 𝜓) → ((𝜑 → 𝜑) → (¬ 𝜒 → ¬ 𝜒))))
3	1, 2	ax-mp 5	. . 3 ⊢ ((((𝜒 → 𝜑) → (¬ 𝜓 → ¬ 𝜓)) → 𝜓) → ((𝜑 → 𝜑) → (¬ 𝜒 → ¬ 𝜒)))
4		meredith 1640	. . 3 ⊢ (((((𝜒 → 𝜑) → (¬ 𝜓 → ¬ 𝜓)) → 𝜓) → ((𝜑 → 𝜑) → (¬ 𝜒 → ¬ 𝜒))) → ((((𝜑 → 𝜑) → (¬ 𝜒 → ¬ 𝜒)) → 𝜒) → (𝜓 → 𝜒)))
5	3, 4	ax-mp 5	. 2 ⊢ ((((𝜑 → 𝜑) → (¬ 𝜒 → ¬ 𝜒)) → 𝜒) → (𝜓 → 𝜒))
6		meredith 1640	. 2 ⊢ (((((𝜑 → 𝜑) → (¬ 𝜒 → ¬ 𝜒)) → 𝜒) → (𝜓 → 𝜒)) → (((𝜓 → 𝜒) → 𝜑) → (𝜒 → 𝜑)))
7	5, 6	ax-mp 5	1 ⊢ (((𝜓 → 𝜒) → 𝜑) → (𝜒 → 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:  merlem4  1644  merlem6  1646