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Metamath Proof Explorer

Theorem merlem7

Description: Between steps 14 and 15 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 22-Dec-2002) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Assertion	merlem7	⊢ ( 𝜑 → ( ( ( 𝜓 → 𝜒 ) → 𝜃 ) → ( ( ( 𝜒 → 𝜏 ) → ( ¬ 𝜃 → ¬ 𝜓 ) ) → 𝜃 ) ) )

Proof

Step	Hyp	Ref	Expression
1		merlem4	⊢ ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → 𝜃 ) → ( ( ( 𝜒 → 𝜏 ) → ( ¬ 𝜃 → ¬ 𝜓 ) ) → 𝜃 ) ) )
2		merlem6	⊢ ( ( ( ( 𝜒 → 𝜏 ) → ( ¬ 𝜃 → ¬ 𝜓 ) ) → 𝜃 ) → ( ( ( ( ( 𝜓 → 𝜒 ) → 𝜃 ) → ( ( ( 𝜒 → 𝜏 ) → ( ¬ 𝜃 → ¬ 𝜓 ) ) → 𝜃 ) ) → ¬ 𝜑 ) → ( ¬ 𝜒 → ¬ 𝜑 ) ) )
3		meredith	⊢ ( ( ( ( ( 𝜒 → 𝜏 ) → ( ¬ 𝜃 → ¬ 𝜓 ) ) → 𝜃 ) → ( ( ( ( ( 𝜓 → 𝜒 ) → 𝜃 ) → ( ( ( 𝜒 → 𝜏 ) → ( ¬ 𝜃 → ¬ 𝜓 ) ) → 𝜃 ) ) → ¬ 𝜑 ) → ( ¬ 𝜒 → ¬ 𝜑 ) ) ) → ( ( ( ( ( ( ( 𝜓 → 𝜒 ) → 𝜃 ) → ( ( ( 𝜒 → 𝜏 ) → ( ¬ 𝜃 → ¬ 𝜓 ) ) → 𝜃 ) ) → ¬ 𝜑 ) → ( ¬ 𝜒 → ¬ 𝜑 ) ) → 𝜒 ) → ( 𝜓 → 𝜒 ) ) )
4	2 3	ax-mp	⊢ ( ( ( ( ( ( ( 𝜓 → 𝜒 ) → 𝜃 ) → ( ( ( 𝜒 → 𝜏 ) → ( ¬ 𝜃 → ¬ 𝜓 ) ) → 𝜃 ) ) → ¬ 𝜑 ) → ( ¬ 𝜒 → ¬ 𝜑 ) ) → 𝜒 ) → ( 𝜓 → 𝜒 ) )
5		meredith	⊢ ( ( ( ( ( ( ( ( 𝜓 → 𝜒 ) → 𝜃 ) → ( ( ( 𝜒 → 𝜏 ) → ( ¬ 𝜃 → ¬ 𝜓 ) ) → 𝜃 ) ) → ¬ 𝜑 ) → ( ¬ 𝜒 → ¬ 𝜑 ) ) → 𝜒 ) → ( 𝜓 → 𝜒 ) ) → ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → 𝜃 ) → ( ( ( 𝜒 → 𝜏 ) → ( ¬ 𝜃 → ¬ 𝜓 ) ) → 𝜃 ) ) ) → ( 𝜑 → ( ( ( 𝜓 → 𝜒 ) → 𝜃 ) → ( ( ( 𝜒 → 𝜏 ) → ( ¬ 𝜃 → ¬ 𝜓 ) ) → 𝜃 ) ) ) ) )
6	4 5	ax-mp	⊢ ( ( ( 𝜓 → 𝜒 ) → ( ( ( 𝜓 → 𝜒 ) → 𝜃 ) → ( ( ( 𝜒 → 𝜏 ) → ( ¬ 𝜃 → ¬ 𝜓 ) ) → 𝜃 ) ) ) → ( 𝜑 → ( ( ( 𝜓 → 𝜒 ) → 𝜃 ) → ( ( ( 𝜒 → 𝜏 ) → ( ¬ 𝜃 → ¬ 𝜓 ) ) → 𝜃 ) ) ) )
7	1 6	ax-mp	⊢ ( 𝜑 → ( ( ( 𝜓 → 𝜒 ) → 𝜃 ) → ( ( ( 𝜒 → 𝜏 ) → ( ¬ 𝜃 → ¬ 𝜓 ) ) → 𝜃 ) ) )