Equivalent to the existential quantifier under negation by

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

which is akin De Morgan’s laws, with \(\forall\) generalizing conjunction.