Theorem merlem3

Description: Step 7 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	merlem3	⊢ ( ( ( 𝜓 → 𝜒 ) → 𝜑 ) → ( 𝜒 → 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		merlem2	⊢ ( ( ( ¬ 𝜒 → ¬ 𝜒 ) → ( ¬ 𝜒 → ¬ 𝜒 ) ) → ( ( 𝜑 → 𝜑 ) → ( ¬ 𝜒 → ¬ 𝜒 ) ) )
2		merlem2	⊢ ( ( ( ( ¬ 𝜒 → ¬ 𝜒 ) → ( ¬ 𝜒 → ¬ 𝜒 ) ) → ( ( 𝜑 → 𝜑 ) → ( ¬ 𝜒 → ¬ 𝜒 ) ) ) → ( ( ( ( 𝜒 → 𝜑 ) → ( ¬ 𝜓 → ¬ 𝜓 ) ) → 𝜓 ) → ( ( 𝜑 → 𝜑 ) → ( ¬ 𝜒 → ¬ 𝜒 ) ) ) )
3	1 2	ax-mp	⊢ ( ( ( ( 𝜒 → 𝜑 ) → ( ¬ 𝜓 → ¬ 𝜓 ) ) → 𝜓 ) → ( ( 𝜑 → 𝜑 ) → ( ¬ 𝜒 → ¬ 𝜒 ) ) )
4		meredith	⊢ ( ( ( ( ( 𝜒 → 𝜑 ) → ( ¬ 𝜓 → ¬ 𝜓 ) ) → 𝜓 ) → ( ( 𝜑 → 𝜑 ) → ( ¬ 𝜒 → ¬ 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ¬ 𝜒 → ¬ 𝜒 ) ) → 𝜒 ) → ( 𝜓 → 𝜒 ) ) )
5	3 4	ax-mp	⊢ ( ( ( ( 𝜑 → 𝜑 ) → ( ¬ 𝜒 → ¬ 𝜒 ) ) → 𝜒 ) → ( 𝜓 → 𝜒 ) )
6		meredith	⊢ ( ( ( ( ( 𝜑 → 𝜑 ) → ( ¬ 𝜒 → ¬ 𝜒 ) ) → 𝜒 ) → ( 𝜓 → 𝜒 ) ) → ( ( ( 𝜓 → 𝜒 ) → 𝜑 ) → ( 𝜒 → 𝜑 ) ) )
7	5 6	ax-mp	⊢ ( ( ( 𝜓 → 𝜒 ) → 𝜑 ) → ( 𝜒 → 𝜑 ) )