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Metamath Proof Explorer

Theorem luklem3

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	luklem3	⊢ ( 𝜑 → ( ( ( ¬ 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜃 → 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		luk-3	⊢ ( 𝜑 → ( ¬ 𝜑 → ¬ 𝜃 ) )
2		luklem2	⊢ ( ( ¬ 𝜑 → ¬ 𝜃 ) → ( ( ( ¬ 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜃 → 𝜒 ) ) )
3	1 2	luklem1	⊢ ( 𝜑 → ( ( ( ¬ 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜃 → 𝜒 ) ) )