Theorem bj-nexdt

Description: Closed form of nexd . (Contributed by BJ, 20-Oct-2019)

	Ref	Expression
Assertion	bj-nexdt	⊢ ( Ⅎ 𝑥 𝜑 → ( ∀ 𝑥 ( 𝜑 → ¬ 𝜓 ) → ( 𝜑 → ¬ ∃ 𝑥 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		nf5r	⊢ ( Ⅎ 𝑥 𝜑 → ( 𝜑 → ∀ 𝑥 𝜑 ) )
2		bj-nexdh	⊢ ( ∀ 𝑥 ( 𝜑 → ¬ 𝜓 ) → ( ( 𝜑 → ∀ 𝑥 𝜑 ) → ( 𝜑 → ¬ ∃ 𝑥 𝜓 ) ) )
3	1 2	syl5com	⊢ ( Ⅎ 𝑥 𝜑 → ( ∀ 𝑥 ( 𝜑 → ¬ 𝜓 ) → ( 𝜑 → ¬ ∃ 𝑥 𝜓 ) ) )