Theorem luklem4

Description: Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	luklem4	⊢ ( ( ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) → 𝜓 ) → 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		luk-2	⊢ ( ( ¬ ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) → ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) ) → ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) )
2		luk-2	⊢ ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 )
3		luklem3	⊢ ( ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) → ( ( ( ¬ ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) → ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) ) → ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) ) → ( ¬ 𝜓 → ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) ) ) )
4	2 3	ax-mp	⊢ ( ( ( ¬ ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) → ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) ) → ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) ) → ( ¬ 𝜓 → ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) ) )
5	1 4	ax-mp	⊢ ( ¬ 𝜓 → ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) )
6		luk-1	⊢ ( ( ¬ 𝜓 → ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) ) → ( ( ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) → 𝜓 ) → ( ¬ 𝜓 → 𝜓 ) ) )
7	5 6	ax-mp	⊢ ( ( ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) → 𝜓 ) → ( ¬ 𝜓 → 𝜓 ) )
8		luk-2	⊢ ( ( ¬ 𝜓 → 𝜓 ) → 𝜓 )
9	7 8	luklem1	⊢ ( ( ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) → 𝜓 ) → 𝜓 )