Metamath Proof Explorer


Theorem merlem7

Description: Between steps 14 and 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 merlem7 ( 𝜑 → ( ( ( 𝜓𝜒 ) → 𝜃 ) → ( ( ( 𝜒𝜏 ) → ( ¬ 𝜃 → ¬ 𝜓 ) ) → 𝜃 ) ) )

Proof

Step Hyp Ref Expression
1 merlem4 ( ( 𝜓𝜒 ) → ( ( ( 𝜓𝜒 ) → 𝜃 ) → ( ( ( 𝜒𝜏 ) → ( ¬ 𝜃 → ¬ 𝜓 ) ) → 𝜃 ) ) )
2 merlem6 ( ( ( ( 𝜒𝜏 ) → ( ¬ 𝜃 → ¬ 𝜓 ) ) → 𝜃 ) → ( ( ( ( ( 𝜓𝜒 ) → 𝜃 ) → ( ( ( 𝜒𝜏 ) → ( ¬ 𝜃 → ¬ 𝜓 ) ) → 𝜃 ) ) → ¬ 𝜑 ) → ( ¬ 𝜒 → ¬ 𝜑 ) ) )
3 meredith ( ( ( ( ( 𝜒𝜏 ) → ( ¬ 𝜃 → ¬ 𝜓 ) ) → 𝜃 ) → ( ( ( ( ( 𝜓𝜒 ) → 𝜃 ) → ( ( ( 𝜒𝜏 ) → ( ¬ 𝜃 → ¬ 𝜓 ) ) → 𝜃 ) ) → ¬ 𝜑 ) → ( ¬ 𝜒 → ¬ 𝜑 ) ) ) → ( ( ( ( ( ( ( 𝜓𝜒 ) → 𝜃 ) → ( ( ( 𝜒𝜏 ) → ( ¬ 𝜃 → ¬ 𝜓 ) ) → 𝜃 ) ) → ¬ 𝜑 ) → ( ¬ 𝜒 → ¬ 𝜑 ) ) → 𝜒 ) → ( 𝜓𝜒 ) ) )
4 2 3 ax-mp ( ( ( ( ( ( ( 𝜓𝜒 ) → 𝜃 ) → ( ( ( 𝜒𝜏 ) → ( ¬ 𝜃 → ¬ 𝜓 ) ) → 𝜃 ) ) → ¬ 𝜑 ) → ( ¬ 𝜒 → ¬ 𝜑 ) ) → 𝜒 ) → ( 𝜓𝜒 ) )
5 meredith ( ( ( ( ( ( ( ( 𝜓𝜒 ) → 𝜃 ) → ( ( ( 𝜒𝜏 ) → ( ¬ 𝜃 → ¬ 𝜓 ) ) → 𝜃 ) ) → ¬ 𝜑 ) → ( ¬ 𝜒 → ¬ 𝜑 ) ) → 𝜒 ) → ( 𝜓𝜒 ) ) → ( ( ( 𝜓𝜒 ) → ( ( ( 𝜓𝜒 ) → 𝜃 ) → ( ( ( 𝜒𝜏 ) → ( ¬ 𝜃 → ¬ 𝜓 ) ) → 𝜃 ) ) ) → ( 𝜑 → ( ( ( 𝜓𝜒 ) → 𝜃 ) → ( ( ( 𝜒𝜏 ) → ( ¬ 𝜃 → ¬ 𝜓 ) ) → 𝜃 ) ) ) ) )
6 4 5 ax-mp ( ( ( 𝜓𝜒 ) → ( ( ( 𝜓𝜒 ) → 𝜃 ) → ( ( ( 𝜒𝜏 ) → ( ¬ 𝜃 → ¬ 𝜓 ) ) → 𝜃 ) ) ) → ( 𝜑 → ( ( ( 𝜓𝜒 ) → 𝜃 ) → ( ( ( 𝜒𝜏 ) → ( ¬ 𝜃 → ¬ 𝜓 ) ) → 𝜃 ) ) ) )
7 1 6 ax-mp ( 𝜑 → ( ( ( 𝜓𝜒 ) → 𝜃 ) → ( ( ( 𝜒𝜏 ) → ( ¬ 𝜃 → ¬ 𝜓 ) ) → 𝜃 ) ) )