Theorem merlem9 1649

Description: Step 18 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
merlem9	⊢ (((𝜑 → 𝜓) → (𝜒 → (𝜃 → (𝜓 → 𝜏)))) → (𝜂 → (𝜒 → (𝜃 → (𝜓 → 𝜏)))))

Proof of Theorem merlem9

Step	Hyp	Ref	Expression
1		merlem6 1646	. . . 4 ⊢ ((𝜃 → (𝜓 → 𝜏)) → (((𝜒 → (𝜃 → (𝜓 → 𝜏))) → ¬ 𝜂) → (¬ 𝜓 → ¬ 𝜂)))
2		merlem8 1648	. . . 4 ⊢ (((𝜃 → (𝜓 → 𝜏)) → (((𝜒 → (𝜃 → (𝜓 → 𝜏))) → ¬ 𝜂) → (¬ 𝜓 → ¬ 𝜂))) → ((((𝜓 → 𝜏) → (¬ (¬ (((𝜒 → (𝜃 → (𝜓 → 𝜏))) → ¬ 𝜂) → (¬ 𝜓 → ¬ 𝜂)) → ¬ 𝜃) → ¬ 𝜑)) → (¬ (((𝜒 → (𝜃 → (𝜓 → 𝜏))) → ¬ 𝜂) → (¬ 𝜓 → ¬ 𝜂)) → ¬ 𝜃)) → (((𝜒 → (𝜃 → (𝜓 → 𝜏))) → ¬ 𝜂) → (¬ 𝜓 → ¬ 𝜂))))
3	1, 2	ax-mp 5	. . 3 ⊢ ((((𝜓 → 𝜏) → (¬ (¬ (((𝜒 → (𝜃 → (𝜓 → 𝜏))) → ¬ 𝜂) → (¬ 𝜓 → ¬ 𝜂)) → ¬ 𝜃) → ¬ 𝜑)) → (¬ (((𝜒 → (𝜃 → (𝜓 → 𝜏))) → ¬ 𝜂) → (¬ 𝜓 → ¬ 𝜂)) → ¬ 𝜃)) → (((𝜒 → (𝜃 → (𝜓 → 𝜏))) → ¬ 𝜂) → (¬ 𝜓 → ¬ 𝜂)))
4		meredith 1640	. . 3 ⊢ (((((𝜓 → 𝜏) → (¬ (¬ (((𝜒 → (𝜃 → (𝜓 → 𝜏))) → ¬ 𝜂) → (¬ 𝜓 → ¬ 𝜂)) → ¬ 𝜃) → ¬ 𝜑)) → (¬ (((𝜒 → (𝜃 → (𝜓 → 𝜏))) → ¬ 𝜂) → (¬ 𝜓 → ¬ 𝜂)) → ¬ 𝜃)) → (((𝜒 → (𝜃 → (𝜓 → 𝜏))) → ¬ 𝜂) → (¬ 𝜓 → ¬ 𝜂))) → (((((𝜒 → (𝜃 → (𝜓 → 𝜏))) → ¬ 𝜂) → (¬ 𝜓 → ¬ 𝜂)) → 𝜓) → (𝜑 → 𝜓)))
5	3, 4	ax-mp 5	. 2 ⊢ (((((𝜒 → (𝜃 → (𝜓 → 𝜏))) → ¬ 𝜂) → (¬ 𝜓 → ¬ 𝜂)) → 𝜓) → (𝜑 → 𝜓))
6		meredith 1640	. 2 ⊢ ((((((𝜒 → (𝜃 → (𝜓 → 𝜏))) → ¬ 𝜂) → (¬ 𝜓 → ¬ 𝜂)) → 𝜓) → (𝜑 → 𝜓)) → (((𝜑 → 𝜓) → (𝜒 → (𝜃 → (𝜓 → 𝜏)))) → (𝜂 → (𝜒 → (𝜃 → (𝜓 → 𝜏))))))
7	5, 6	ax-mp 5	1 ⊢ (((𝜑 → 𝜓) → (𝜒 → (𝜃 → (𝜓 → 𝜏)))) → (𝜂 → (𝜒 → (𝜃 → (𝜓 → 𝜏)))))

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:  merlem10  1650