Let $\mathcal{L}=\{\sim\}$ and $\Sigma_\infty$ be the set of axioms stating that:
(i) $\sim$ is an equivalence relation
(ii) Every equivalence class is infinite
(iii) there are infinitely many equivalence classes
It suffices to show that every $\mathcal{L}$ formula $\exists y\varphi(x,y)$ is $\Sigma_\infty$-equivalent to a quantifier free $\psi(x)$, where $x=(x_1,\ldots,x_n)$ are distinct variables, and $\varphi(x,y)$ is a conjunction of literals. Since it's fairly trivial when $\varphi(x,y)$ has a conjunct of the form $x_i=y$, we can assume we need to reduce $$\exists y (\bigwedge_{i\in I} x_i\not=y\wedge\bigwedge_{j\in J}x_j\sim y\wedge\bigwedge_{h\in H}\neg (x_j\sim y))$$ My question is whether or not the following condition holds:
$$\Sigma_\infty\vdash\exists y (\bigwedge_{i\in I} x_i\not=y\wedge\bigwedge_{j\in J}x_j\sim y\wedge\bigwedge_{h\in H}\neg (x_j\sim y))\leftrightarrow(\bigwedge_{j\in J,h\in H}\neg(x_i\sim x_j)\wedge\bigwedge_{j,j'\in J}x_j\sim x_{j'})$$ I can't think of a reason why it would not, but I'm a bit worried about ignoring the information about non-identical elements in the unreduced formula.