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Metamath Proof Explorer

Theorem merlem12

Description: Step 28 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 merlem12 ( ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → 𝜑 ) → 𝜑 )

Proof

Step Hyp Ref Expression
1 merlem5 ⊢ ( ( 𝜒 → 𝜒 ) → ( ¬ ¬ 𝜒 → 𝜒 ) )
2 merlem2 ⊢ ( ( ( 𝜒 → 𝜒 ) → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) )
3 1 2 ax-mp ⊢ ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) )
4 merlem4 ⊢ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ( ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → 𝜑 ) → ( ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → 𝜑 ) → 𝜑 ) ) )
5 3 4 ax-mp ⊢ ( ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → 𝜑 ) → ( ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → 𝜑 ) → 𝜑 ) )
6 merlem11 ⊢ ( ( ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → 𝜑 ) → ( ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → 𝜑 ) → 𝜑 ) ) → ( ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → 𝜑 ) → 𝜑 ) )
7 5 6 ax-mp ⊢ ( ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → 𝜑 ) → 𝜑 )