Negated Existential Quantifier of a conjunction

Well, indeed, you have correctly applied quantifier duality and deMorgan's Law.

After that you may also the apply Conditional Equivalence (aka Implication Equivalence).   Which is that: $(\neg p\vee r)$ and $(p\to r)$ are equivalent for any statements $p,r$ $ \tiny\text{(in classical logic)}$.

$$\begin{align}&\neg(\exists x)~(P(x)\wedge Q(x))\\&(\forall x)~\neg(P(x)\wedge Q(x))&&\raise{2ex}\text{Existential/Universal Duality}\\&(\forall x)~(\neg P(x)\vee\neg Q(x))&&\raise{2ex}\text{DeMorgan's Rule}\\&(\forall x)~(P(x)\to\neg Q(x))&&\raise{2ex}\text{Conditional Equivalence}\end{align}$$

All these statements are considered equivalent.

The last is preferred only because it has the familiar form of "All $P$ are not $Q$," which is the natural language negation of "Some $P$ is $Q$."

That is all.

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PS: Do note, however, that is not what you wrote. The negation is only on the consequent ($Q$) not the whole conditional.