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Metamath Proof Explorer

Theorem luklem7

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	luklem7	⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		luk-1	⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( ( 𝜓 → 𝜒 ) → 𝜒 ) → ( 𝜑 → 𝜒 ) ) )
2		luklem5	⊢ ( 𝜓 → ( ( 𝜓 → 𝜒 ) → 𝜓 ) )
3		luk-1	⊢ ( ( ( 𝜓 → 𝜒 ) → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( ( 𝜓 → 𝜒 ) → 𝜒 ) ) )
4	2 3	luklem1	⊢ ( 𝜓 → ( ( 𝜓 → 𝜒 ) → ( ( 𝜓 → 𝜒 ) → 𝜒 ) ) )
5		luklem6	⊢ ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜓 → 𝜒 ) → 𝜒 ) ) → ( ( 𝜓 → 𝜒 ) → 𝜒 ) )
6	4 5	luklem1	⊢ ( 𝜓 → ( ( 𝜓 → 𝜒 ) → 𝜒 ) )
7		luk-1	⊢ ( ( 𝜓 → ( ( 𝜓 → 𝜒 ) → 𝜒 ) ) → ( ( ( ( 𝜓 → 𝜒 ) → 𝜒 ) → ( 𝜑 → 𝜒 ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) )
8	6 7	ax-mp	⊢ ( ( ( ( 𝜓 → 𝜒 ) → 𝜒 ) → ( 𝜑 → 𝜒 ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) )
9	1 8	luklem1	⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) )