Theorem merlem5

Description: Step 11 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	merlem5	⊢ ( ( 𝜑 → 𝜓 ) → ( ¬ ¬ 𝜑 → 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		meredith	⊢ ( ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜓 → ¬ 𝜓 ) ) → 𝜓 ) → 𝜓 ) → ( ( 𝜓 → 𝜓 ) → ( 𝜓 → 𝜓 ) ) )
2		meredith	⊢ ( ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜓 → ¬ ¬ ¬ 𝜑 ) ) → 𝜓 ) → 𝜑 ) → ( ( 𝜑 → 𝜓 ) → ( ¬ ¬ 𝜑 → 𝜓 ) ) )
3		merlem1	⊢ ( ( ( ( 𝜑 → 𝜓 ) → ( ¬ ¬ 𝜑 → 𝜓 ) ) → ¬ ( ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜓 → ¬ 𝜓 ) ) → 𝜓 ) → 𝜓 ) → ( ( 𝜓 → 𝜓 ) → ( 𝜓 → 𝜓 ) ) ) ) → ( ¬ 𝜑 → ¬ ( ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜓 → ¬ 𝜓 ) ) → 𝜓 ) → 𝜓 ) → ( ( 𝜓 → 𝜓 ) → ( 𝜓 → 𝜓 ) ) ) ) )
4		merlem4	⊢ ( ( ( ( ( 𝜑 → 𝜓 ) → ( ¬ ¬ 𝜑 → 𝜓 ) ) → ¬ ( ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜓 → ¬ 𝜓 ) ) → 𝜓 ) → 𝜓 ) → ( ( 𝜓 → 𝜓 ) → ( 𝜓 → 𝜓 ) ) ) ) → ( ¬ 𝜑 → ¬ ( ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜓 → ¬ 𝜓 ) ) → 𝜓 ) → 𝜓 ) → ( ( 𝜓 → 𝜓 ) → ( 𝜓 → 𝜓 ) ) ) ) ) → ( ( ( ( ( ( 𝜑 → 𝜓 ) → ( ¬ ¬ 𝜑 → 𝜓 ) ) → ¬ ( ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜓 → ¬ 𝜓 ) ) → 𝜓 ) → 𝜓 ) → ( ( 𝜓 → 𝜓 ) → ( 𝜓 → 𝜓 ) ) ) ) → ( ¬ 𝜑 → ¬ ( ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜓 → ¬ 𝜓 ) ) → 𝜓 ) → 𝜓 ) → ( ( 𝜓 → 𝜓 ) → ( 𝜓 → 𝜓 ) ) ) ) ) → 𝜑 ) → ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜓 → ¬ ¬ ¬ 𝜑 ) ) → 𝜓 ) → 𝜑 ) ) )
5	3 4	ax-mp	⊢ ( ( ( ( ( ( 𝜑 → 𝜓 ) → ( ¬ ¬ 𝜑 → 𝜓 ) ) → ¬ ( ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜓 → ¬ 𝜓 ) ) → 𝜓 ) → 𝜓 ) → ( ( 𝜓 → 𝜓 ) → ( 𝜓 → 𝜓 ) ) ) ) → ( ¬ 𝜑 → ¬ ( ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜓 → ¬ 𝜓 ) ) → 𝜓 ) → 𝜓 ) → ( ( 𝜓 → 𝜓 ) → ( 𝜓 → 𝜓 ) ) ) ) ) → 𝜑 ) → ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜓 → ¬ ¬ ¬ 𝜑 ) ) → 𝜓 ) → 𝜑 ) )
6		meredith	⊢ ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → ( ¬ ¬ 𝜑 → 𝜓 ) ) → ¬ ( ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜓 → ¬ 𝜓 ) ) → 𝜓 ) → 𝜓 ) → ( ( 𝜓 → 𝜓 ) → ( 𝜓 → 𝜓 ) ) ) ) → ( ¬ 𝜑 → ¬ ( ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜓 → ¬ 𝜓 ) ) → 𝜓 ) → 𝜓 ) → ( ( 𝜓 → 𝜓 ) → ( 𝜓 → 𝜓 ) ) ) ) ) → 𝜑 ) → ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜓 → ¬ ¬ ¬ 𝜑 ) ) → 𝜓 ) → 𝜑 ) ) → ( ( ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜓 → ¬ ¬ ¬ 𝜑 ) ) → 𝜓 ) → 𝜑 ) → ( ( 𝜑 → 𝜓 ) → ( ¬ ¬ 𝜑 → 𝜓 ) ) ) → ( ( ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜓 → ¬ 𝜓 ) ) → 𝜓 ) → 𝜓 ) → ( ( 𝜓 → 𝜓 ) → ( 𝜓 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ¬ ¬ 𝜑 → 𝜓 ) ) ) ) )
7	5 6	ax-mp	⊢ ( ( ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜓 → ¬ ¬ ¬ 𝜑 ) ) → 𝜓 ) → 𝜑 ) → ( ( 𝜑 → 𝜓 ) → ( ¬ ¬ 𝜑 → 𝜓 ) ) ) → ( ( ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜓 → ¬ 𝜓 ) ) → 𝜓 ) → 𝜓 ) → ( ( 𝜓 → 𝜓 ) → ( 𝜓 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ¬ ¬ 𝜑 → 𝜓 ) ) ) )
8	2 7	ax-mp	⊢ ( ( ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜓 → ¬ 𝜓 ) ) → 𝜓 ) → 𝜓 ) → ( ( 𝜓 → 𝜓 ) → ( 𝜓 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ¬ ¬ 𝜑 → 𝜓 ) ) )
9	1 8	ax-mp	⊢ ( ( 𝜑 → 𝜓 ) → ( ¬ ¬ 𝜑 → 𝜓 ) )