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 )