Theorem nexdv

Description: Deduction for generalization rule for negated wff. (Contributed by NM, 5-Aug-1993) Reduce dependencies on axioms. (Revised by Wolf Lammen, 13-Jul-2020) (Proof shortened by Wolf Lammen, 10-Oct-2021)

	Ref	Expression
Hypothesis	nexdv.1	⊢ ( 𝜑 → ¬ 𝜓 )
Assertion	nexdv	⊢ ( 𝜑 → ¬ ∃ 𝑥 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		nexdv.1	⊢ ( 𝜑 → ¬ 𝜓 )
2		ax-5	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
3	2 1	nexdh	⊢ ( 𝜑 → ¬ ∃ 𝑥 𝜓 )