Theorem nexd 2257

Description: Deduction for generalization rule for negated wff. (Contributed by Mario Carneiro, 24-Sep-2016.)

Hypotheses

Ref	Expression
nexd.1	⊢ Ⅎ𝑥𝜑
nexd.2	⊢ (𝜑 → ¬ 𝜓)

Assertion

Ref	Expression
nexd	⊢ (𝜑 → ¬ ∃𝑥𝜓)

Proof of Theorem nexd

Step	Hyp	Ref	Expression
1		nexd.1	. . 3 ⊢ Ⅎ𝑥𝜑
2	1	nf5ri 2231	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
3		nexd.2	. 2 ⊢ (𝜑 → ¬ 𝜓)
4	2, 3	nexdh 1897	1 ⊢ (𝜑 → ¬ ∃𝑥𝜓)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∃wex 1811  Ⅎwnf 1815

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-12 2212

This proof depends on definitions:  df-bi 210  df-ex 1812  df-nf 1816

This theorem is used by:  axrepnd  10686  axunnd  10688