Theorem nex 1457

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 1456	. 2 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
2		nex.1	. 2 ⊢ ¬ 𝜑
3	1, 2	mpgbi 1409	1 ⊢ ¬ ∃𝑥𝜑

Syntax hints:  ¬ wn 3  ∃wex 1449

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-5 1404  ax-gen 1406  ax-ie2 1451

This theorem depends on definitions:  df-bi 116  df-tru 1315  df-fal 1318

This theorem is referenced by:  ru  2875  0nelxp  4525  0xp  4577  dm0  4711  co02  5008  0fv  5408  mpo0  5793  0npr  7232