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Metamath Proof Explorer

Theorem merlem9

Description: Step 18 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 merlem9 ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) ) ) → ( 𝜂 → ( 𝜒 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) ) ) )

Proof

Step Hyp Ref Expression
1 merlem6 ⊢ ( ( 𝜃 → ( 𝜓 → 𝜏 ) ) → ( ( ( 𝜒 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) ) → ¬ 𝜂 ) → ( ¬ 𝜓 → ¬ 𝜂 ) ) )
2 merlem8 ⊢ ( ( ( 𝜃 → ( 𝜓 → 𝜏 ) ) → ( ( ( 𝜒 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) ) → ¬ 𝜂 ) → ( ¬ 𝜓 → ¬ 𝜂 ) ) ) → ( ( ( ( 𝜓 → 𝜏 ) → ( ¬ ( ¬ ( ( ( 𝜒 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) ) → ¬ 𝜂 ) → ( ¬ 𝜓 → ¬ 𝜂 ) ) → ¬ 𝜃 ) → ¬ 𝜑 ) ) → ( ¬ ( ( ( 𝜒 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) ) → ¬ 𝜂 ) → ( ¬ 𝜓 → ¬ 𝜂 ) ) → ¬ 𝜃 ) ) → ( ( ( 𝜒 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) ) → ¬ 𝜂 ) → ( ¬ 𝜓 → ¬ 𝜂 ) ) ) )
3 1 2 ax-mp ⊢ ( ( ( ( 𝜓 → 𝜏 ) → ( ¬ ( ¬ ( ( ( 𝜒 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) ) → ¬ 𝜂 ) → ( ¬ 𝜓 → ¬ 𝜂 ) ) → ¬ 𝜃 ) → ¬ 𝜑 ) ) → ( ¬ ( ( ( 𝜒 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) ) → ¬ 𝜂 ) → ( ¬ 𝜓 → ¬ 𝜂 ) ) → ¬ 𝜃 ) ) → ( ( ( 𝜒 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) ) → ¬ 𝜂 ) → ( ¬ 𝜓 → ¬ 𝜂 ) ) )
4 meredith ⊢ ( ( ( ( ( 𝜓 → 𝜏 ) → ( ¬ ( ¬ ( ( ( 𝜒 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) ) → ¬ 𝜂 ) → ( ¬ 𝜓 → ¬ 𝜂 ) ) → ¬ 𝜃 ) → ¬ 𝜑 ) ) → ( ¬ ( ( ( 𝜒 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) ) → ¬ 𝜂 ) → ( ¬ 𝜓 → ¬ 𝜂 ) ) → ¬ 𝜃 ) ) → ( ( ( 𝜒 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) ) → ¬ 𝜂 ) → ( ¬ 𝜓 → ¬ 𝜂 ) ) ) → ( ( ( ( ( 𝜒 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) ) → ¬ 𝜂 ) → ( ¬ 𝜓 → ¬ 𝜂 ) ) → 𝜓 ) → ( 𝜑 → 𝜓 ) ) )
5 3 4 ax-mp ⊢ ( ( ( ( ( 𝜒 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) ) → ¬ 𝜂 ) → ( ¬ 𝜓 → ¬ 𝜂 ) ) → 𝜓 ) → ( 𝜑 → 𝜓 ) )
6 meredith ⊢ ( ( ( ( ( ( 𝜒 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) ) → ¬ 𝜂 ) → ( ¬ 𝜓 → ¬ 𝜂 ) ) → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) ) ) → ( 𝜂 → ( 𝜒 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) ) ) ) )
7 5 6 ax-mp ⊢ ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) ) ) → ( 𝜂 → ( 𝜒 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) ) ) )