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Relation to universal quantification

Equivalent to the universal quantifier under negation by

\[\begin{align*}
  \exists x \> \lnot P(x) \>
  & \equiv \> \lnot \forall x \> P(x)
  \\
  \lnot \exists x \> P(x) \>
  & \equiv \> \forall x \> \lnot P(x).
\end{align*}
\]

which is akin De Morgan’s laws, with \(\exists\) generalizing disjunction.