Theorem nex 1555

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 1543	. 2 ⊢ (∀x ¬ φ ↔ ¬ ∃xφ)
2		nex.1	. 2 ⊢ ¬ φ
3	1, 2	mpgbi 1549	1 ⊢ ¬ ∃xφ

Syntax hints:  ¬ wn 3  ∃wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546

This theorem depends on definitions:  df-bi 177  df-ex 1542

This theorem is referenced by:  ru  3046  0nel1c  4160  xp0r  4843  dm0  4919  co02  5093  co01  5094  nenpw1pwlem2  6086