Theorem nex 1430

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 1429	. 2 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
2		nex.1	. 2 ⊢ ¬ 𝜑
3	1, 2	mpgbi 1382	1 ⊢ ¬ ∃𝑥𝜑

Syntax hints:  ¬ wn 3  ∃wex 1422

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-5 1377  ax-gen 1379  ax-ie2 1424

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291

This theorem is referenced by:  ru  2825  0nelxp  4428  0xp  4476  dm0  4608  co02  4898  0fv  5284  mpt20  5653  0npr  6945