(Caveat: I am new to descriptive set theory so I suspect the question may have a straightforward answer.)
Let $X$ be a Polish space. We say a Borel equivalence relation $E\subset X\times X$ is smooth if there exists a countable family of Borel sets $\{A_i\}_{i\geq 1}$ such that $xEy$ just in case for every $i$, $x\in A_i\Leftrightarrow y\in A_i$.
We know that there exist non-smooth equivalence relations of $[0,1]$, the Vitali equivalence relation being one example ($xEy$ iff $x-y\in\mathbb{Q}$). What I am curious about is whether the existence of a non-smooth equivalence relation of $[0,1]$ requires the Axiom of Choice (in particular, whether in the Solovay model, every Borel equivalence relation of the unit interval is smooth). I am suspecting that it does, given that the existence of non-measurable set requires AC, and non-smoothness seems related to non-measurableness (vaguely). But I wasn't able to find any references and didn't know how to proceed myself. Any guidance would be greatly appreciated.