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 ) )