Is there any reason to prefer one phrasing over the other? The following two sentences in the language of $\mathbb{N}$ are logically equivalent, in the sense that first-order logic alone is enough to get from one to the other.


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*For all $a,b;$ if there exists $k$ such that $ak=b$, then $a\mid b$.

*For all $a,b$ and all $k$, if $ak=b$, then $a\mid b$.


Many similar examples abound in mathematics. Is there any reason to prefer one phrasing over the other?

The first variant seems to occur more naturally and match the ways we think as it compares two predicates $\Phi(a,b) \equiv \exists k(ak=b)$ and $\Psi(a,b)\equiv a|b$. It also has the advantage that we can in fact formulate the equivalence
$$ \forall a\forall b(\exists k(ak=b)\leftrightarrow a|b).$$
On the other hand, the second variant has the advantage that all quantors have been "pulled out". Moreover, there is no existential quantifier so that we may infer
$$ ak=b\to a|b$$
for arbitrary values of $k,a,b$, which can be advantageous for formal deductions.
