Theorem nexdh

Description: Deduction for generalization rule for negated wff. (Contributed by NM, 2-Jan-2002)

	Ref	Expression
Hypotheses	nexdh.1	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
	nexdh.2	⊢ ( 𝜑 → ¬ 𝜓 )
Assertion	nexdh	⊢ ( 𝜑 → ¬ ∃ 𝑥 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		nexdh.1	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
2		nexdh.2	⊢ ( 𝜑 → ¬ 𝜓 )
3	1 2	alrimih	⊢ ( 𝜑 → ∀ 𝑥 ¬ 𝜓 )
4		alnex	⊢ ( ∀ 𝑥 ¬ 𝜓 ↔ ¬ ∃ 𝑥 𝜓 )
5	3 4	sylib	⊢ ( 𝜑 → ¬ ∃ 𝑥 𝜓 )