Want to show Quantifier elimination and completeness of this set of axioms... 
Let $\Sigma_\infty$ be a set of axioms in the language $\{\sim\}$ (where $\sim$ is a binary relation
symbol) that states:
(i) $\sim$ is an equivalence relation;
(ii) every equivalence class is infinite;
(iii) there are infinitely many equivalence classes.
Show that $\Sigma_{\infty}$ admits QE and is complete. (It is given that it is also possible to use Vaught's test
to prove completeness.)

I think I have shown that $\Sigma_\infty$ admits QE, but am not sure
how to show completeness. There is a theorem, however, that states that if a set of sentences $\Sigma$ has a model and admits QE, and there exists an $L$-structure that can be embedded in every model of $\Sigma$, then $\Sigma$ is complete.
Thanks.

Your quantifier elimination procedure will transform any sentence in the language into an equivalent quantifier-free formula whose truth value can be calculated (as either true or false).
