Theorem luk-2 1664

Description: 2 of 3 axioms for propositional calculus due to Lukasiewicz, derived from Meredith's sole axiom. (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
luk-2	⊢ ((¬ 𝜑 → 𝜑) → 𝜑)

Proof of Theorem luk-2

Step	Hyp	Ref	Expression
1		merlem5 1654	. . . . 5 ⊢ ((𝜑 → ¬ (¬ 𝜑 → 𝜑)) → (¬ ¬ 𝜑 → ¬ (¬ 𝜑 → 𝜑)))
2		merlem4 1653	. . . . 5 ⊢ (((𝜑 → ¬ (¬ 𝜑 → 𝜑)) → (¬ ¬ 𝜑 → ¬ (¬ 𝜑 → 𝜑))) → ((((𝜑 → ¬ (¬ 𝜑 → 𝜑)) → (¬ ¬ 𝜑 → ¬ (¬ 𝜑 → 𝜑))) → ¬ 𝜑) → ((((𝜑 → ¬ (¬ 𝜑 → 𝜑)) → (¬ ¬ 𝜑 → ¬ (¬ 𝜑 → 𝜑))) → ¬ 𝜑) → ¬ 𝜑)))
3	1, 2	ax-mp 5	. . . 4 ⊢ ((((𝜑 → ¬ (¬ 𝜑 → 𝜑)) → (¬ ¬ 𝜑 → ¬ (¬ 𝜑 → 𝜑))) → ¬ 𝜑) → ((((𝜑 → ¬ (¬ 𝜑 → 𝜑)) → (¬ ¬ 𝜑 → ¬ (¬ 𝜑 → 𝜑))) → ¬ 𝜑) → ¬ 𝜑))
4		merlem11 1660	. . . 4 ⊢ (((((𝜑 → ¬ (¬ 𝜑 → 𝜑)) → (¬ ¬ 𝜑 → ¬ (¬ 𝜑 → 𝜑))) → ¬ 𝜑) → ((((𝜑 → ¬ (¬ 𝜑 → 𝜑)) → (¬ ¬ 𝜑 → ¬ (¬ 𝜑 → 𝜑))) → ¬ 𝜑) → ¬ 𝜑)) → ((((𝜑 → ¬ (¬ 𝜑 → 𝜑)) → (¬ ¬ 𝜑 → ¬ (¬ 𝜑 → 𝜑))) → ¬ 𝜑) → ¬ 𝜑))
5	3, 4	ax-mp 5	. . 3 ⊢ ((((𝜑 → ¬ (¬ 𝜑 → 𝜑)) → (¬ ¬ 𝜑 → ¬ (¬ 𝜑 → 𝜑))) → ¬ 𝜑) → ¬ 𝜑)
6		meredith 1649	. . 3 ⊢ (((((𝜑 → ¬ (¬ 𝜑 → 𝜑)) → (¬ ¬ 𝜑 → ¬ (¬ 𝜑 → 𝜑))) → ¬ 𝜑) → ¬ 𝜑) → ((¬ 𝜑 → 𝜑) → ((¬ 𝜑 → 𝜑) → 𝜑)))
7	5, 6	ax-mp 5	. 2 ⊢ ((¬ 𝜑 → 𝜑) → ((¬ 𝜑 → 𝜑) → 𝜑))
8		merlem11 1660	. 2 ⊢ (((¬ 𝜑 → 𝜑) → ((¬ 𝜑 → 𝜑) → 𝜑)) → ((¬ 𝜑 → 𝜑) → 𝜑))
9	7, 8	ax-mp 5	1 ⊢ ((¬ 𝜑 → 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:  luklem4  1669