Theorem merlem10

Description: Step 19 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	merlem10	⊢ ( ( 𝜑 → ( 𝜑 → 𝜓 ) ) → ( 𝜃 → ( 𝜑 → 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		meredith	⊢ ( ( ( ( ( 𝜑 → 𝜑 ) → ( ¬ 𝜑 → ¬ 𝜑 ) ) → 𝜑 ) → 𝜑 ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → 𝜑 ) ) )
2		meredith	⊢ ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ( ¬ 𝜑 → ¬ 𝜃 ) ) → 𝜑 ) → 𝜑 ) → ( ( 𝜑 → ( 𝜑 → 𝜓 ) ) → ( 𝜃 → ( 𝜑 → 𝜓 ) ) ) )
3		merlem9	⊢ ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ( ¬ 𝜑 → ¬ 𝜃 ) ) → 𝜑 ) → 𝜑 ) → ( ( 𝜑 → ( 𝜑 → 𝜓 ) ) → ( 𝜃 → ( 𝜑 → 𝜓 ) ) ) ) → ( ( ( ( ( ( 𝜑 → 𝜑 ) → ( ¬ 𝜑 → ¬ 𝜑 ) ) → 𝜑 ) → 𝜑 ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → 𝜑 ) ) ) → ( ( 𝜑 → ( 𝜑 → 𝜓 ) ) → ( 𝜃 → ( 𝜑 → 𝜓 ) ) ) ) )
4	2 3	ax-mp	⊢ ( ( ( ( ( ( 𝜑 → 𝜑 ) → ( ¬ 𝜑 → ¬ 𝜑 ) ) → 𝜑 ) → 𝜑 ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → 𝜑 ) ) ) → ( ( 𝜑 → ( 𝜑 → 𝜓 ) ) → ( 𝜃 → ( 𝜑 → 𝜓 ) ) ) )
5	1 4	ax-mp	⊢ ( ( 𝜑 → ( 𝜑 → 𝜓 ) ) → ( 𝜃 → ( 𝜑 → 𝜓 ) ) )