Theorem nex 1548

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 1547	. 2 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
2		nex.1	. 2 ⊢ ¬ 𝜑
3	1, 2	mpgbi 1500	1 ⊢ ¬ ∃𝑥𝜑

Syntax hints:  ¬ wn 3  ∃wex 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-5 1495  ax-gen 1497  ax-ie2 1542

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-fal 1403

This theorem is referenced by:  ru  3030  0nelxp  4753  0xp  4806  dm0  4945  co02  5250  0fv  5677  mpo0  6090  0npr  7702  0g0  13458  gsum0g  13478