Past exam of the model theory course of the University of Oxford / 2011 / 2 / b / ii / Solution

A structure embedding preserves and reflects all atomic formulas, hence all quantifier-free formulas by induction on Boolean combinations. If $T$ has quantifier elimination, replace any $\phi (\mathbf{x})$ by a quantifier-free $\theta (\mathbf{x})$ equivalent over $T$. For $\mathcal{A}\subseteq \mathcal{B}\models T$ and a tuple $\mathbf{a}$ from $A$,Thus $\mathcal{A}⪯\mathcal{B}$, and the preceding criterion proves $\boxed{\text{quantifier elimination implies model completeness}.}$