Help solving a negate and simplify problem

Question

just wondering if anyone could help with the answer to this so I can check my working.

"negate and simplify the following"

$$∀x∃y∈C∃z[∃w(P→Q)∨¬D]$$

Answer 1

$\exists x P(x)($Exists $xPx)$ also read as at least one $xP(x)$ or $\exists^{\ge1}P(x)$

$\forall x \neg P(x)($forall $x\neg P(x))$ 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 $\forall x \neg P(x)$,

Based on this we have:

$$\neg(∀x∃y∈C∃z∃w(P→Q)∨¬D)$$

$$\equiv\exists x\forall y\in C\forall z\forall w P\land \neg Q\land D$$

If $P,Q,D$ are not about $x,y,z,w$ then:

$$\equiv P\land\neg Q\land D$$

This simplified the statement.