Theorem nexdv 1955

Description: Deduction for generalization rule for negated wff. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
nexdv.1	⊢ (𝜑 → ¬ 𝜓)

Assertion

Ref	Expression
nexdv	⊢ (𝜑 → ¬ ∃𝑥𝜓)

Distinct variable group:   𝜑,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem nexdv

Step	Hyp	Ref	Expression
1		ax-17 1540	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
2		nexdv.1	. 2 ⊢ (𝜑 → ¬ 𝜓)
3	1, 2	nexd 1627	1 ⊢ (𝜑 → ¬ ∃𝑥𝜓)

Syntax hints:  ¬ wn 3   → wi 4  ∃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  ax-17 1540

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

This theorem is referenced by:  lgsquadlem3  15320  pw1nct  15647