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 → ( 𝜓𝜒 ) ) → ( 𝜓 → ( 𝜑𝜒 ) ) )

Proof

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