Metamath Proof Explorer


Theorem luk-2

Description: 2 of 3 axioms for propositional calculus due to Lukasiewicz, derived from Meredith's sole axiom. (Contributed by NM, 14-Dec-2002) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion luk-2 ( ( ¬ 𝜑𝜑 ) → 𝜑 )

Proof

Step Hyp Ref Expression
1 merlem5 ( ( 𝜑 → ¬ ( ¬ 𝜑𝜑 ) ) → ( ¬ ¬ 𝜑 → ¬ ( ¬ 𝜑𝜑 ) ) )
2 merlem4 ( ( ( 𝜑 → ¬ ( ¬ 𝜑𝜑 ) ) → ( ¬ ¬ 𝜑 → ¬ ( ¬ 𝜑𝜑 ) ) ) → ( ( ( ( 𝜑 → ¬ ( ¬ 𝜑𝜑 ) ) → ( ¬ ¬ 𝜑 → ¬ ( ¬ 𝜑𝜑 ) ) ) → ¬ 𝜑 ) → ( ( ( ( 𝜑 → ¬ ( ¬ 𝜑𝜑 ) ) → ( ¬ ¬ 𝜑 → ¬ ( ¬ 𝜑𝜑 ) ) ) → ¬ 𝜑 ) → ¬ 𝜑 ) ) )
3 1 2 ax-mp ( ( ( ( 𝜑 → ¬ ( ¬ 𝜑𝜑 ) ) → ( ¬ ¬ 𝜑 → ¬ ( ¬ 𝜑𝜑 ) ) ) → ¬ 𝜑 ) → ( ( ( ( 𝜑 → ¬ ( ¬ 𝜑𝜑 ) ) → ( ¬ ¬ 𝜑 → ¬ ( ¬ 𝜑𝜑 ) ) ) → ¬ 𝜑 ) → ¬ 𝜑 ) )
4 merlem11 ( ( ( ( ( 𝜑 → ¬ ( ¬ 𝜑𝜑 ) ) → ( ¬ ¬ 𝜑 → ¬ ( ¬ 𝜑𝜑 ) ) ) → ¬ 𝜑 ) → ( ( ( ( 𝜑 → ¬ ( ¬ 𝜑𝜑 ) ) → ( ¬ ¬ 𝜑 → ¬ ( ¬ 𝜑𝜑 ) ) ) → ¬ 𝜑 ) → ¬ 𝜑 ) ) → ( ( ( ( 𝜑 → ¬ ( ¬ 𝜑𝜑 ) ) → ( ¬ ¬ 𝜑 → ¬ ( ¬ 𝜑𝜑 ) ) ) → ¬ 𝜑 ) → ¬ 𝜑 ) )
5 3 4 ax-mp ( ( ( ( 𝜑 → ¬ ( ¬ 𝜑𝜑 ) ) → ( ¬ ¬ 𝜑 → ¬ ( ¬ 𝜑𝜑 ) ) ) → ¬ 𝜑 ) → ¬ 𝜑 )
6 meredith ( ( ( ( ( 𝜑 → ¬ ( ¬ 𝜑𝜑 ) ) → ( ¬ ¬ 𝜑 → ¬ ( ¬ 𝜑𝜑 ) ) ) → ¬ 𝜑 ) → ¬ 𝜑 ) → ( ( ¬ 𝜑𝜑 ) → ( ( ¬ 𝜑𝜑 ) → 𝜑 ) ) )
7 5 6 ax-mp ( ( ¬ 𝜑𝜑 ) → ( ( ¬ 𝜑𝜑 ) → 𝜑 ) )
8 merlem11 ( ( ( ¬ 𝜑𝜑 ) → ( ( ¬ 𝜑𝜑 ) → 𝜑 ) ) → ( ( ¬ 𝜑𝜑 ) → 𝜑 ) )
9 7 8 ax-mp ( ( ¬ 𝜑𝜑 ) → 𝜑 )