Simplifying theories with quantifier elimination Let $\Sigma$ be a theory that has quantifier elimination. I'm trying to show that there is then an equivalent theory $\Sigma^*$, with each $\sigma\in\Sigma^*$ of the form $\forall x\psi(x)$ or $\forall x\exists y\varphi(x,y)$ ($\varphi, \psi$ being quantifier free).
I'm not quite sure how to get started. 
 A: The deleted answer by Henning Makholm had essentially the right idea, modulo a few details. Let me spell it out a bit more.
Let $\Sigma^*$ consist of all consequences of $\Sigma$ which are either quantifier-free sentences or which have the form $\forall \overline{x}\,(\exists y\, \psi(\overline{x},y))\leftrightarrow \theta(\overline{x})$, where $\overline{x}$ is a tuple of variables (possibly empty), and $\psi$ and $\theta$ are quantifier-free formulas. Such a sentence is $\Pi_2$ ($\forall\exists$) with a single existential quantifier when put in prenex normal form. 
Now clearly $\Sigma\models \Sigma^*$. To show $\Sigma^*\models \Sigma$, you need to prove by induction on quantifier complexity (working from the innermost quantifier out) that $\Sigma^*$ also has quantifier elimination, and hence that every sentence in $\Sigma$ is equivalent to one of the quantifier-free consequences of $\Sigma$ included in $\Sigma^*$.
Let me remark that I like to include the symbols $\top$ and $\bot$ in first-order logic. If you don't include logical symbols for true and false, and if your language has no constant symbols, you will run into the problem that there are no quantifier-free sentences. In this case, you need to also include all of the 1-quantifier consequences of $\Sigma$ ($\forall x\, \varphi(x)$ and $\exists x\, \varphi(x)$) in $\Sigma^*$, and show that modulo $\Sigma^*$, every sentence is equivalent to a $1$-quantifier sentence. This is probably why the statement of your problem included the form $\forall x\, \varphi(x)$ instead of the form $\varphi$ where $\varphi$ is a quantifier-free sentence.
In the comments above, you've indicated that you're looking for a theory axiomatized by sentences of the form $\forall x\, \exists y\, \varphi(x,y)$, where $x$ and $y$ are single variables. Just to reiterate the point I made in the comments, such sentences won't suffice in general. It's a very rare property that a theory can be axiomatized by sentences with at most two quantifiers.
