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Metamath Proof Explorer

Theorem merlem13

Description: Step 35 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 merlem13 ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) → 𝜓 ) )

Proof

Step Hyp Ref Expression
1 merlem12 ⊢ ( ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) )
2 merlem12 ⊢ ( ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) → ¬ ¬ 𝜑 )
3 merlem5 ⊢ ( ( ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) → ¬ ¬ 𝜑 ) → ( ¬ ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) → ¬ ¬ 𝜑 ) )
4 2 3 ax-mp ⊢ ( ¬ ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) → ¬ ¬ 𝜑 )
5 merlem6 ⊢ ( ( ¬ ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) → ¬ ¬ 𝜑 ) → ( ( ( ( ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) → 𝜓 ) → ( ¬ ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) → ¬ ¬ 𝜑 ) ) → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) → ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) ) )
6 4 5 ax-mp ⊢ ( ( ( ( ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) → 𝜓 ) → ( ¬ ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) → ¬ ¬ 𝜑 ) ) → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) → ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) )
7 meredith ⊢ ( ( ( ( ( ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) → 𝜓 ) → ( ¬ ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) → ¬ ¬ 𝜑 ) ) → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) → ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) ) → ( ( ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) → ( ¬ 𝜑 → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) ) )
8 6 7 ax-mp ⊢ ( ( ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) → ( ¬ 𝜑 → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) )
9 1 8 ax-mp ⊢ ( ¬ 𝜑 → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) )
10 merlem6 ⊢ ( ( ¬ 𝜑 → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) → ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜑 → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) ) → 𝜑 ) → ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜑 → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) ) → 𝜑 ) → 𝜑 ) ) )
11 9 10 ax-mp ⊢ ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜑 → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) ) → 𝜑 ) → ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜑 → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) ) → 𝜑 ) → 𝜑 ) )
12 merlem11 ⊢ ( ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜑 → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) ) → 𝜑 ) → ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜑 → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) ) → 𝜑 ) → 𝜑 ) ) → ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜑 → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) ) → 𝜑 ) → 𝜑 ) )
13 11 12 ax-mp ⊢ ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜑 → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) ) → 𝜑 ) → 𝜑 )
14 meredith ⊢ ( ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜑 → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) ) → 𝜑 ) → 𝜑 ) → ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) → 𝜓 ) ) )
15 13 14 ax-mp ⊢ ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) → 𝜓 ) )