Theorem nexdv 1996

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 1579	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
2		nexdv.1	. 2 ⊢ (𝜑 → ¬ 𝜓)
3	1, 2	nexd 1666	1 ⊢ (𝜑 → ¬ ∃𝑥𝜓)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∃wex 1545

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-5 1500  ax-gen 1502  ax-ie2 1547  ax-17 1579

This proof depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408

This theorem is used by:  lgsquadlem3  16198  umgredgnlp  16393  pw1nct  17033