Theorem merlem1

Description: Step 3 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (The step numbers refer to Meredith's original paper.) (Contributed by NM, 14-Dec-2002) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	merlem1	⊢ ( ( ( 𝜒 → ( ¬ 𝜑 → 𝜓 ) ) → 𝜏 ) → ( 𝜑 → 𝜏 ) )

Proof

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