Theorem nexd 1627

Description: Deduction for generalization rule for negated wff. (Contributed by NM, 2-Jan-2002.)

Hypotheses

Ref	Expression
nexd.1	⊢ (𝜑 → ∀𝑥𝜑)
nexd.2	⊢ (𝜑 → ¬ 𝜓)

Assertion

Ref	Expression
nexd	⊢ (𝜑 → ¬ ∃𝑥𝜓)

Proof of Theorem nexd

Step	Hyp	Ref	Expression
1		nexd.1	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
2		nexd.2	. . 3 ⊢ (𝜑 → ¬ 𝜓)
3	1, 2	alrimih 1483	. 2 ⊢ (𝜑 → ∀𝑥 ¬ 𝜓)
4		alnex 1513	. 2 ⊢ (∀𝑥 ¬ 𝜓 ↔ ¬ ∃𝑥𝜓)
5	3, 4	sylib 122	1 ⊢ (𝜑 → ¬ ∃𝑥𝜓)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1362  ∃wex 1506

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 615  ax-in2 616  ax-5 1461  ax-gen 1463  ax-ie2 1508

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370

This theorem is referenced by:  nexdv  1955