Theorem merlem13 1654

Description: Step 35 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
merlem13	⊢ ((𝜑 → 𝜓) → (((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ¬ 𝜑) → 𝜓))

Proof of Theorem merlem13

Step	Hyp	Ref	Expression
1		merlem12 1653	. . . . 5 ⊢ (((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ¬ 𝜑)) → ¬ ((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ¬ 𝜑))
2		merlem12 1653	. . . . . . . 8 ⊢ (((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ¬ 𝜑) → ¬ ¬ 𝜑)
3		merlem5 1646	. . . . . . . 8 ⊢ ((((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ¬ 𝜑) → ¬ ¬ 𝜑) → (¬ ¬ ((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ¬ 𝜑) → ¬ ¬ 𝜑))
4	2, 3	ax-mp 5	. . . . . . 7 ⊢ (¬ ¬ ((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ¬ 𝜑) → ¬ ¬ 𝜑)
5		merlem6 1647	. . . . . . 7 ⊢ ((¬ ¬ ((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ¬ 𝜑) → ¬ ¬ 𝜑) → ((((¬ ((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ¬ 𝜑) → 𝜓) → (¬ ¬ ((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ¬ 𝜑) → ¬ ¬ 𝜑)) → ¬ ((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ¬ 𝜑)) → ((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ¬ 𝜑))))
6	4, 5	ax-mp 5	. . . . . 6 ⊢ ((((¬ ((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ¬ 𝜑) → 𝜓) → (¬ ¬ ((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ¬ 𝜑) → ¬ ¬ 𝜑)) → ¬ ((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ¬ 𝜑)) → ((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ¬ 𝜑)))
7		meredith 1641	. . . . . 6 ⊢ (((((¬ ((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ¬ 𝜑) → 𝜓) → (¬ ¬ ((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ¬ 𝜑) → ¬ ¬ 𝜑)) → ¬ ((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ¬ 𝜑)) → ((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ¬ 𝜑))) → ((((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ¬ 𝜑)) → ¬ ((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ¬ 𝜑)) → (¬ 𝜑 → ¬ ((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ¬ 𝜑))))
8	6, 7	ax-mp 5	. . . . 5 ⊢ ((((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ¬ 𝜑)) → ¬ ((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ¬ 𝜑)) → (¬ 𝜑 → ¬ ((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ¬ 𝜑)))
9	1, 8	ax-mp 5	. . . 4 ⊢ (¬ 𝜑 → ¬ ((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ¬ 𝜑))
10		merlem6 1647	. . . 4 ⊢ ((¬ 𝜑 → ¬ ((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ¬ 𝜑)) → ((((𝜓 → 𝜓) → (¬ 𝜑 → ¬ ((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ¬ 𝜑))) → 𝜑) → ((((𝜓 → 𝜓) → (¬ 𝜑 → ¬ ((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ¬ 𝜑))) → 𝜑) → 𝜑)))
11	9, 10	ax-mp 5	. . 3 ⊢ ((((𝜓 → 𝜓) → (¬ 𝜑 → ¬ ((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ¬ 𝜑))) → 𝜑) → ((((𝜓 → 𝜓) → (¬ 𝜑 → ¬ ((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ¬ 𝜑))) → 𝜑) → 𝜑))
12		merlem11 1652	. . 3 ⊢ (((((𝜓 → 𝜓) → (¬ 𝜑 → ¬ ((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ¬ 𝜑))) → 𝜑) → ((((𝜓 → 𝜓) → (¬ 𝜑 → ¬ ((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ¬ 𝜑))) → 𝜑) → 𝜑)) → ((((𝜓 → 𝜓) → (¬ 𝜑 → ¬ ((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ¬ 𝜑))) → 𝜑) → 𝜑))
13	11, 12	ax-mp 5	. 2 ⊢ ((((𝜓 → 𝜓) → (¬ 𝜑 → ¬ ((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ¬ 𝜑))) → 𝜑) → 𝜑)
14		meredith 1641	. 2 ⊢ (((((𝜓 → 𝜓) → (¬ 𝜑 → ¬ ((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ¬ 𝜑))) → 𝜑) → 𝜑) → ((𝜑 → 𝜓) → (((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ¬ 𝜑) → 𝜓)))
15	13, 14	ax-mp 5	1 ⊢ ((𝜑 → 𝜓) → (((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ¬ 𝜑) → 𝜓))

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:  luk-1  1655