Theorem nex 1493

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 1492	. 2 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
2		nex.1	. 2 ⊢ ¬ 𝜑
3	1, 2	mpgbi 1445	1 ⊢ ¬ ∃𝑥𝜑

Syntax hints:  ¬ wn 3  ∃wex 1485

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 609  ax-in2 610  ax-5 1440  ax-gen 1442  ax-ie2 1487

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354

This theorem is referenced by:  ru  2954  0nelxp  4639  0xp  4691  dm0  4825  co02  5124  0fv  5531  mpo0  5923  0npr  7445  0g0  12630