Theorem merlem7 1645

Description: Between steps 14 and 15 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
merlem7	⊢ (𝜑 → (((𝜓 → 𝜒) → 𝜃) → (((𝜒 → 𝜏) → (¬ 𝜃 → ¬ 𝜓)) → 𝜃)))

Proof of Theorem merlem7

Step	Hyp	Ref	Expression
1		merlem4 1642	. 2 ⊢ ((𝜓 → 𝜒) → (((𝜓 → 𝜒) → 𝜃) → (((𝜒 → 𝜏) → (¬ 𝜃 → ¬ 𝜓)) → 𝜃)))
2		merlem6 1644	. . . 4 ⊢ ((((𝜒 → 𝜏) → (¬ 𝜃 → ¬ 𝜓)) → 𝜃) → (((((𝜓 → 𝜒) → 𝜃) → (((𝜒 → 𝜏) → (¬ 𝜃 → ¬ 𝜓)) → 𝜃)) → ¬ 𝜑) → (¬ 𝜒 → ¬ 𝜑)))
3		meredith 1638	. . . 4 ⊢ (((((𝜒 → 𝜏) → (¬ 𝜃 → ¬ 𝜓)) → 𝜃) → (((((𝜓 → 𝜒) → 𝜃) → (((𝜒 → 𝜏) → (¬ 𝜃 → ¬ 𝜓)) → 𝜃)) → ¬ 𝜑) → (¬ 𝜒 → ¬ 𝜑))) → (((((((𝜓 → 𝜒) → 𝜃) → (((𝜒 → 𝜏) → (¬ 𝜃 → ¬ 𝜓)) → 𝜃)) → ¬ 𝜑) → (¬ 𝜒 → ¬ 𝜑)) → 𝜒) → (𝜓 → 𝜒)))
4	2, 3	ax-mp 5	. . 3 ⊢ (((((((𝜓 → 𝜒) → 𝜃) → (((𝜒 → 𝜏) → (¬ 𝜃 → ¬ 𝜓)) → 𝜃)) → ¬ 𝜑) → (¬ 𝜒 → ¬ 𝜑)) → 𝜒) → (𝜓 → 𝜒))
5		meredith 1638	. . 3 ⊢ ((((((((𝜓 → 𝜒) → 𝜃) → (((𝜒 → 𝜏) → (¬ 𝜃 → ¬ 𝜓)) → 𝜃)) → ¬ 𝜑) → (¬ 𝜒 → ¬ 𝜑)) → 𝜒) → (𝜓 → 𝜒)) → (((𝜓 → 𝜒) → (((𝜓 → 𝜒) → 𝜃) → (((𝜒 → 𝜏) → (¬ 𝜃 → ¬ 𝜓)) → 𝜃))) → (𝜑 → (((𝜓 → 𝜒) → 𝜃) → (((𝜒 → 𝜏) → (¬ 𝜃 → ¬ 𝜓)) → 𝜃)))))
6	4, 5	ax-mp 5	. 2 ⊢ (((𝜓 → 𝜒) → (((𝜓 → 𝜒) → 𝜃) → (((𝜒 → 𝜏) → (¬ 𝜃 → ¬ 𝜓)) → 𝜃))) → (𝜑 → (((𝜓 → 𝜒) → 𝜃) → (((𝜒 → 𝜏) → (¬ 𝜃 → ¬ 𝜓)) → 𝜃))))
7	1, 6	ax-mp 5	1 ⊢ (𝜑 → (((𝜓 → 𝜒) → 𝜃) → (((𝜒 → 𝜏) → (¬ 𝜃 → ¬ 𝜓)) → 𝜃)))

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:  merlem8  1646