$\exists x P(x)($Exists $xPx)$ also read as at least one $xP(x)$ or $\exists^{\ge1}P(x)$
$\forall x \
eg P(x)($forall $x\
eg P(x))$ also read as at most zero $xP(x)$, or $\exists^{<1}xP(x)\equiv\exists^{\le0}xP(x)$
It's same as does not exist any $x$ that $P(x)$,
So the negation of $\exists x P(x)$ is $\forall x \
eg P(x)$,
Based on this we have:
$$\
eg(∀x∃y∈C∃z∃w(P→Q)∨¬D)$$
$$\equiv\exists x\forall y\in C\forall z\forall w P\land \
eg Q\land D$$
If $P,Q,D$ are not about $x,y,z,w$ then:
$$\equiv P\land\
eg Q\land D$$
This simplified the statement.