I'm following along an example that takes a formula and converts it into Skolem Normal Form.
In one of the steps, the formula goes $\text{from: }\forall x~P(x) ~\wedge~ \exists x~\forall y~\exists z~\neg Q(x,y,z)\\\quad\text{to: }\exists x~\forall y~\exists z~(P(x)\wedge\neg Q(x,y,z))$
Why did the existential quantifier take preference over the universal quantifier? Aren't they two different scopes and then ∃x,x be replaced by another variable?
That is indeed really strange. In fact, not just strange, but these are not equivalent: If you have two objects $a$ and $b$ in your domain, and only $a$ has property $p$, and nothing stands in the $Q$ relationship, then $\exists x\forall y\exists z(Px\land\neg Qxyz)$ is true but $ \forall xPx\land\exists x\forall y\exists z\ \neg Qxyz$ is false.
So, the latter statement is not the former statement in some Skolem normal form.