Theorem bj-nexdh

Description: Closed form of nexdh (actually, its general instance). (Contributed by BJ, 6-May-2019)

	Ref	Expression
Assertion	bj-nexdh	⊢ ( ∀ 𝑥 ( 𝜑 → ¬ 𝜓 ) → ( ( 𝜒 → ∀ 𝑥 𝜑 ) → ( 𝜒 → ¬ ∃ 𝑥 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		sylgt	⊢ ( ∀ 𝑥 ( 𝜑 → ¬ 𝜓 ) → ( ( 𝜒 → ∀ 𝑥 𝜑 ) → ( 𝜒 → ∀ 𝑥 ¬ 𝜓 ) ) )
2		alnex	⊢ ( ∀ 𝑥 ¬ 𝜓 ↔ ¬ ∃ 𝑥 𝜓 )
3	1 2	syl8ib	⊢ ( ∀ 𝑥 ( 𝜑 → ¬ 𝜓 ) → ( ( 𝜒 → ∀ 𝑥 𝜑 ) → ( 𝜒 → ¬ ∃ 𝑥 𝜓 ) ) )