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Metamath Proof Explorer

Theorem merlem5

Description: Step 11 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 14-Dec-2002) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion merlem5 ( ( 𝜑 → 𝜓 ) → ( ¬ ¬ 𝜑 → 𝜓 ) )

Proof

Step Hyp Ref Expression
1 meredith ⊢ ( ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜓 → ¬ 𝜓 ) ) → 𝜓 ) → 𝜓 ) → ( ( 𝜓 → 𝜓 ) → ( 𝜓 → 𝜓 ) ) )
2 meredith ⊢ ( ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜓 → ¬ ¬ ¬ 𝜑 ) ) → 𝜓 ) → 𝜑 ) → ( ( 𝜑 → 𝜓 ) → ( ¬ ¬ 𝜑 → 𝜓 ) ) )
3 merlem1 ⊢ ( ( ( ( 𝜑 → 𝜓 ) → ( ¬ ¬ 𝜑 → 𝜓 ) ) → ¬ ( ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜓 → ¬ 𝜓 ) ) → 𝜓 ) → 𝜓 ) → ( ( 𝜓 → 𝜓 ) → ( 𝜓 → 𝜓 ) ) ) ) → ( ¬ 𝜑 → ¬ ( ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜓 → ¬ 𝜓 ) ) → 𝜓 ) → 𝜓 ) → ( ( 𝜓 → 𝜓 ) → ( 𝜓 → 𝜓 ) ) ) ) )
4 merlem4 ⊢ ( ( ( ( ( 𝜑 → 𝜓 ) → ( ¬ ¬ 𝜑 → 𝜓 ) ) → ¬ ( ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜓 → ¬ 𝜓 ) ) → 𝜓 ) → 𝜓 ) → ( ( 𝜓 → 𝜓 ) → ( 𝜓 → 𝜓 ) ) ) ) → ( ¬ 𝜑 → ¬ ( ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜓 → ¬ 𝜓 ) ) → 𝜓 ) → 𝜓 ) → ( ( 𝜓 → 𝜓 ) → ( 𝜓 → 𝜓 ) ) ) ) ) → ( ( ( ( ( ( 𝜑 → 𝜓 ) → ( ¬ ¬ 𝜑 → 𝜓 ) ) → ¬ ( ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜓 → ¬ 𝜓 ) ) → 𝜓 ) → 𝜓 ) → ( ( 𝜓 → 𝜓 ) → ( 𝜓 → 𝜓 ) ) ) ) → ( ¬ 𝜑 → ¬ ( ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜓 → ¬ 𝜓 ) ) → 𝜓 ) → 𝜓 ) → ( ( 𝜓 → 𝜓 ) → ( 𝜓 → 𝜓 ) ) ) ) ) → 𝜑 ) → ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜓 → ¬ ¬ ¬ 𝜑 ) ) → 𝜓 ) → 𝜑 ) ) )
5 3 4 ax-mp ⊢ ( ( ( ( ( ( 𝜑 → 𝜓 ) → ( ¬ ¬ 𝜑 → 𝜓 ) ) → ¬ ( ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜓 → ¬ 𝜓 ) ) → 𝜓 ) → 𝜓 ) → ( ( 𝜓 → 𝜓 ) → ( 𝜓 → 𝜓 ) ) ) ) → ( ¬ 𝜑 → ¬ ( ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜓 → ¬ 𝜓 ) ) → 𝜓 ) → 𝜓 ) → ( ( 𝜓 → 𝜓 ) → ( 𝜓 → 𝜓 ) ) ) ) ) → 𝜑 ) → ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜓 → ¬ ¬ ¬ 𝜑 ) ) → 𝜓 ) → 𝜑 ) )
6 meredith ⊢ ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → ( ¬ ¬ 𝜑 → 𝜓 ) ) → ¬ ( ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜓 → ¬ 𝜓 ) ) → 𝜓 ) → 𝜓 ) → ( ( 𝜓 → 𝜓 ) → ( 𝜓 → 𝜓 ) ) ) ) → ( ¬ 𝜑 → ¬ ( ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜓 → ¬ 𝜓 ) ) → 𝜓 ) → 𝜓 ) → ( ( 𝜓 → 𝜓 ) → ( 𝜓 → 𝜓 ) ) ) ) ) → 𝜑 ) → ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜓 → ¬ ¬ ¬ 𝜑 ) ) → 𝜓 ) → 𝜑 ) ) → ( ( ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜓 → ¬ ¬ ¬ 𝜑 ) ) → 𝜓 ) → 𝜑 ) → ( ( 𝜑 → 𝜓 ) → ( ¬ ¬ 𝜑 → 𝜓 ) ) ) → ( ( ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜓 → ¬ 𝜓 ) ) → 𝜓 ) → 𝜓 ) → ( ( 𝜓 → 𝜓 ) → ( 𝜓 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ¬ ¬ 𝜑 → 𝜓 ) ) ) ) )
7 5 6 ax-mp ⊢ ( ( ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜓 → ¬ ¬ ¬ 𝜑 ) ) → 𝜓 ) → 𝜑 ) → ( ( 𝜑 → 𝜓 ) → ( ¬ ¬ 𝜑 → 𝜓 ) ) ) → ( ( ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜓 → ¬ 𝜓 ) ) → 𝜓 ) → 𝜓 ) → ( ( 𝜓 → 𝜓 ) → ( 𝜓 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ¬ ¬ 𝜑 → 𝜓 ) ) ) )
8 2 7 ax-mp ⊢ ( ( ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜓 → ¬ 𝜓 ) ) → 𝜓 ) → 𝜓 ) → ( ( 𝜓 → 𝜓 ) → ( 𝜓 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ¬ ¬ 𝜑 → 𝜓 ) ) )
9 1 8 ax-mp ⊢ ( ( 𝜑 → 𝜓 ) → ( ¬ ¬ 𝜑 → 𝜓 ) )