Theorem luk-2

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.)

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

Proof

Step	Hyp	Ref	Expression
1		merlem5	⊢ ( ( 𝜑 → ¬ ( ¬ 𝜑 → 𝜑 ) ) → ( ¬ ¬ 𝜑 → ¬ ( ¬ 𝜑 → 𝜑 ) ) )
2		merlem4	⊢ ( ( ( 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	⊢ ( ( ( ( 𝜑 → ¬ ( ¬ 𝜑 → 𝜑 ) ) → ( ¬ ¬ 𝜑 → ¬ ( ¬ 𝜑 → 𝜑 ) ) ) → ¬ 𝜑 ) → ( ( ( ( 𝜑 → ¬ ( ¬ 𝜑 → 𝜑 ) ) → ( ¬ ¬ 𝜑 → ¬ ( ¬ 𝜑 → 𝜑 ) ) ) → ¬ 𝜑 ) → ¬ 𝜑 ) )
4		merlem11	⊢ ( ( ( ( ( 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	⊢ ( ( ( ( 𝜑 → ¬ ( ¬ 𝜑 → 𝜑 ) ) → ( ¬ ¬ 𝜑 → ¬ ( ¬ 𝜑 → 𝜑 ) ) ) → ¬ 𝜑 ) → ¬ 𝜑 )
6		meredith	⊢ ( ( ( ( ( 𝜑 → ¬ ( ¬ 𝜑 → 𝜑 ) ) → ( ¬ ¬ 𝜑 → ¬ ( ¬ 𝜑 → 𝜑 ) ) ) → ¬ 𝜑 ) → ¬ 𝜑 ) → ( ( ¬ 𝜑 → 𝜑 ) → ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) ) )
7	5 6	ax-mp	⊢ ( ( ¬ 𝜑 → 𝜑 ) → ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) )
8		merlem11	⊢ ( ( ( ¬ 𝜑 → 𝜑 ) → ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) ) → ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) )
9	7 8	ax-mp	⊢ ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 )