Metamath Proof Explorer


Theorem luklem2

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 luklem2 ( ( 𝜑 → ¬ 𝜓 ) → ( ( ( 𝜑𝜒 ) → 𝜃 ) → ( 𝜓𝜃 ) ) )

Proof

Step Hyp Ref Expression
1 luk-1 ( ( 𝜑 → ¬ 𝜓 ) → ( ( ¬ 𝜓𝜒 ) → ( 𝜑𝜒 ) ) )
2 luk-3 ( 𝜓 → ( ¬ 𝜓𝜒 ) )
3 luk-1 ( ( 𝜓 → ( ¬ 𝜓𝜒 ) ) → ( ( ( ¬ 𝜓𝜒 ) → ( 𝜑𝜒 ) ) → ( 𝜓 → ( 𝜑𝜒 ) ) ) )
4 2 3 ax-mp ( ( ( ¬ 𝜓𝜒 ) → ( 𝜑𝜒 ) ) → ( 𝜓 → ( 𝜑𝜒 ) ) )
5 1 4 luklem1 ( ( 𝜑 → ¬ 𝜓 ) → ( 𝜓 → ( 𝜑𝜒 ) ) )
6 luk-1 ( ( 𝜓 → ( 𝜑𝜒 ) ) → ( ( ( 𝜑𝜒 ) → 𝜃 ) → ( 𝜓𝜃 ) ) )
7 5 6 luklem1 ( ( 𝜑 → ¬ 𝜓 ) → ( ( ( 𝜑𝜒 ) → 𝜃 ) → ( 𝜓𝜃 ) ) )