So I am studying the theory of quantifer elimination and have come across the following equivalence. I am defining quantifier elimination as: a theory $T$ admits quantifier-elimination (QE) if every formula $\phi$ is equivalent to a quantifier-free one.
I then want to prove that $T$ admits QE if and only if for any quantifier-free formula $\phi(x,y_1,y_2,...,y_n)$ of $L$, there exists a quantifier-free formula $\psi(y_1,...,y_n)$ such that $$T \models (\forall y_1,...,y_n)(\psi \iff(\exists x)\phi)$$.
I have proved the forward direction I think:
Suppose $T$ admits QE, and suppose $\phi(x,y_1,y_2,...,y_n)$ is a quantifier-free formula. Consider $\exists x\phi(x,y_1,y_2,...,y_n)$. Then as $T$ admits QE, $T \models (\forall y_1,...,y_n)(\psi \iff(\exists x)\phi)$, where $\psi$ is quantifier free formula. I think this is simply by definitions. However, I do not know how to prove the converse, so any help would be hugely appreciated. Thank you.