Theorem merlem4

Description: Step 8 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	merlem4	⊢ ( 𝜏 → ( ( 𝜏 → 𝜑 ) → ( 𝜃 → 𝜑 ) ) )

Proof

Step	Hyp	Ref	Expression
1		meredith	⊢ ( ( ( ( ( 𝜑 → 𝜑 ) → ( ¬ 𝜃 → ¬ 𝜃 ) ) → 𝜃 ) → 𝜏 ) → ( ( 𝜏 → 𝜑 ) → ( 𝜃 → 𝜑 ) ) )
2		merlem3	⊢ ( ( ( ( ( ( 𝜑 → 𝜑 ) → ( ¬ 𝜃 → ¬ 𝜃 ) ) → 𝜃 ) → 𝜏 ) → ( ( 𝜏 → 𝜑 ) → ( 𝜃 → 𝜑 ) ) ) → ( 𝜏 → ( ( 𝜏 → 𝜑 ) → ( 𝜃 → 𝜑 ) ) ) )
3	1 2	ax-mp	⊢ ( 𝜏 → ( ( 𝜏 → 𝜑 ) → ( 𝜃 → 𝜑 ) ) )