Theorem nex 1480

Description: Generalization rule for negated wff. (Contributed by NM, 18-May-1994.)

Hypothesis

Ref	Expression
nex.1	⊢ ¬ 𝜑

Assertion

Ref	Expression
nex	⊢ ¬ ∃𝑥𝜑

Proof of Theorem nex

Step	Hyp	Ref	Expression
1		alnex 1479	. 2 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
2		nex.1	. 2 ⊢ ¬ 𝜑
3	1, 2	mpgbi 1432	1 ⊢ ¬ ∃𝑥𝜑

Syntax hints:  ¬ wn 3  ∃wex 1472

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-5 1427  ax-gen 1429  ax-ie2 1474

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-fal 1341

This theorem is referenced by:  ru  2936  0nelxp  4615  0xp  4667  dm0  4801  co02  5100  0fv  5504  mpo0  5892  0npr  7404