Theorem merlem6

Description: Step 12 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	merlem6	⊢ ( 𝜒 → ( ( ( 𝜓 → 𝜒 ) → 𝜑 ) → ( 𝜃 → 𝜑 ) ) )

Proof

Step	Hyp	Ref	Expression
1		merlem4	⊢ ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → 𝜑 ) → ( 𝜃 → 𝜑 ) ) )
2		merlem3	⊢ ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → 𝜑 ) → ( 𝜃 → 𝜑 ) ) ) → ( 𝜒 → ( ( ( 𝜓 → 𝜒 ) → 𝜑 ) → ( 𝜃 → 𝜑 ) ) ) )
3	1 2	ax-mp	⊢ ( 𝜒 → ( ( ( 𝜓 → 𝜒 ) → 𝜑 ) → ( 𝜃 → 𝜑 ) ) )