Metamath Proof Explorer


Theorem merlem8

Description: Step 15 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 22-Dec-2002) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion merlem8 ( ( ( 𝜓𝜒 ) → 𝜃 ) → ( ( ( 𝜒𝜏 ) → ( ¬ 𝜃 → ¬ 𝜓 ) ) → 𝜃 ) )

Proof

Step Hyp Ref Expression
1 meredith ( ( ( ( ( 𝜑𝜑 ) → ( ¬ 𝜑 → ¬ 𝜑 ) ) → 𝜑 ) → 𝜑 ) → ( ( 𝜑𝜑 ) → ( 𝜑𝜑 ) ) )
2 merlem7 ( ( ( ( ( ( 𝜑𝜑 ) → ( ¬ 𝜑 → ¬ 𝜑 ) ) → 𝜑 ) → 𝜑 ) → ( ( 𝜑𝜑 ) → ( 𝜑𝜑 ) ) ) → ( ( ( 𝜓𝜒 ) → 𝜃 ) → ( ( ( 𝜒𝜏 ) → ( ¬ 𝜃 → ¬ 𝜓 ) ) → 𝜃 ) ) )
3 1 2 ax-mp ( ( ( 𝜓𝜒 ) → 𝜃 ) → ( ( ( 𝜒𝜏 ) → ( ¬ 𝜃 → ¬ 𝜓 ) ) → 𝜃 ) )