Theorem luk-2 1421

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 1411	. . . . 5 ⊢ ((φ → ¬ (¬ φ → φ)) → (¬ ¬ φ → ¬ (¬ φ → φ)))
2		merlem4 1410	. . . . 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 1417	. . . 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		ax-meredith 1406	. . 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 1417	. 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-meredith 1406

This theorem is referenced by:  luklem4  1426