Suppose $Pxy$ is a $2$-ary predicate and $\phi$ is just some fixed arbitrary predicate form.
$\forall x [ (\exists y Pxy) \longrightarrow \phi ]$
$\forall x \exists y [Pxy \longrightarrow \phi]$
Is the only difference between 1. and 2. in the case where $\exists y$ is not true? i.e. We may be in an interpretation $\mathcal{I}$ where the universe/domain of discourse may be empty, so does that then make 1. true and 2. false?
So it seems to me that if we restrict ourselves to interpretations $\mathcal{I}$ whose domain of discourse is not empty then 1. and 2. are logically equivalent$^1$.
Remarks: $^1$I'm thinking of the interpretations in the context of first-order logic, but what about second-order logic or other higher orders of logic? (Though I have not studied anything beyond first order logic, but maybe someone can say something about this? But this probably is another question in its own right)