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.