Theorem luk-1

Description: 1 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-1	⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		meredith	⊢ ( ( ( ( ( 𝜒 → 𝜒 ) → ( ¬ ¬ ¬ 𝜑 → ¬ 𝜑 ) ) → ¬ ¬ 𝜑 ) → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) )
2		merlem13	⊢ ( ( 𝜑 → 𝜓 ) → ( ( ( ( 𝜒 → 𝜒 ) → ( ¬ ¬ ¬ 𝜑 → ¬ 𝜑 ) ) → ¬ ¬ 𝜑 ) → 𝜓 ) )
3		merlem13	⊢ ( ( ( 𝜑 → 𝜓 ) → ( ( ( ( 𝜒 → 𝜒 ) → ( ¬ ¬ ¬ 𝜑 → ¬ 𝜑 ) ) → ¬ ¬ 𝜑 ) → 𝜓 ) ) → ( ( ( ( ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) → 𝜑 ) → ( ¬ ¬ ¬ ( 𝜑 → 𝜓 ) → ¬ ( 𝜑 → 𝜓 ) ) ) → ¬ ¬ ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜒 → 𝜒 ) → ( ¬ ¬ ¬ 𝜑 → ¬ 𝜑 ) ) → ¬ ¬ 𝜑 ) → 𝜓 ) ) )
4	2 3	ax-mp	⊢ ( ( ( ( ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) → 𝜑 ) → ( ¬ ¬ ¬ ( 𝜑 → 𝜓 ) → ¬ ( 𝜑 → 𝜓 ) ) ) → ¬ ¬ ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜒 → 𝜒 ) → ( ¬ ¬ ¬ 𝜑 → ¬ 𝜑 ) ) → ¬ ¬ 𝜑 ) → 𝜓 ) )
5		meredith	⊢ ( ( ( ( ( ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) → 𝜑 ) → ( ¬ ¬ ¬ ( 𝜑 → 𝜓 ) → ¬ ( 𝜑 → 𝜓 ) ) ) → ¬ ¬ ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜒 → 𝜒 ) → ( ¬ ¬ ¬ 𝜑 → ¬ 𝜑 ) ) → ¬ ¬ 𝜑 ) → 𝜓 ) ) → ( ( ( ( ( ( 𝜒 → 𝜒 ) → ( ¬ ¬ ¬ 𝜑 → ¬ 𝜑 ) ) → ¬ ¬ 𝜑 ) → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) )
6	4 5	ax-mp	⊢ ( ( ( ( ( ( 𝜒 → 𝜒 ) → ( ¬ ¬ ¬ 𝜑 → ¬ 𝜑 ) ) → ¬ ¬ 𝜑 ) → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) )
7	1 6	ax-mp	⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) )