Theorem merlem9

Description: Step 18 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 22-Dec-2002) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	merlem9	⊢ ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) ) ) → ( 𝜂 → ( 𝜒 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		merlem6	⊢ ( ( 𝜃 → ( 𝜓 → 𝜏 ) ) → ( ( ( 𝜒 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) ) → ¬ 𝜂 ) → ( ¬ 𝜓 → ¬ 𝜂 ) ) )
2		merlem8	⊢ ( ( ( 𝜃 → ( 𝜓 → 𝜏 ) ) → ( ( ( 𝜒 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) ) → ¬ 𝜂 ) → ( ¬ 𝜓 → ¬ 𝜂 ) ) ) → ( ( ( ( 𝜓 → 𝜏 ) → ( ¬ ( ¬ ( ( ( 𝜒 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) ) → ¬ 𝜂 ) → ( ¬ 𝜓 → ¬ 𝜂 ) ) → ¬ 𝜃 ) → ¬ 𝜑 ) ) → ( ¬ ( ( ( 𝜒 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) ) → ¬ 𝜂 ) → ( ¬ 𝜓 → ¬ 𝜂 ) ) → ¬ 𝜃 ) ) → ( ( ( 𝜒 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) ) → ¬ 𝜂 ) → ( ¬ 𝜓 → ¬ 𝜂 ) ) ) )
3	1 2	ax-mp	⊢ ( ( ( ( 𝜓 → 𝜏 ) → ( ¬ ( ¬ ( ( ( 𝜒 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) ) → ¬ 𝜂 ) → ( ¬ 𝜓 → ¬ 𝜂 ) ) → ¬ 𝜃 ) → ¬ 𝜑 ) ) → ( ¬ ( ( ( 𝜒 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) ) → ¬ 𝜂 ) → ( ¬ 𝜓 → ¬ 𝜂 ) ) → ¬ 𝜃 ) ) → ( ( ( 𝜒 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) ) → ¬ 𝜂 ) → ( ¬ 𝜓 → ¬ 𝜂 ) ) )
4		meredith	⊢ ( ( ( ( ( 𝜓 → 𝜏 ) → ( ¬ ( ¬ ( ( ( 𝜒 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) ) → ¬ 𝜂 ) → ( ¬ 𝜓 → ¬ 𝜂 ) ) → ¬ 𝜃 ) → ¬ 𝜑 ) ) → ( ¬ ( ( ( 𝜒 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) ) → ¬ 𝜂 ) → ( ¬ 𝜓 → ¬ 𝜂 ) ) → ¬ 𝜃 ) ) → ( ( ( 𝜒 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) ) → ¬ 𝜂 ) → ( ¬ 𝜓 → ¬ 𝜂 ) ) ) → ( ( ( ( ( 𝜒 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) ) → ¬ 𝜂 ) → ( ¬ 𝜓 → ¬ 𝜂 ) ) → 𝜓 ) → ( 𝜑 → 𝜓 ) ) )
5	3 4	ax-mp	⊢ ( ( ( ( ( 𝜒 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) ) → ¬ 𝜂 ) → ( ¬ 𝜓 → ¬ 𝜂 ) ) → 𝜓 ) → ( 𝜑 → 𝜓 ) )
6		meredith	⊢ ( ( ( ( ( ( 𝜒 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) ) → ¬ 𝜂 ) → ( ¬ 𝜓 → ¬ 𝜂 ) ) → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) ) ) → ( 𝜂 → ( 𝜒 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) ) ) ) )
7	5 6	ax-mp	⊢ ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) ) ) → ( 𝜂 → ( 𝜒 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) ) ) )