= Solution
Assume $T$ has <quantifier elimination>, and let $j:\mathcal M\to\mathcal N$ be a <structure embedding> between its models. For any <first-order formula> $\varphi(\bar x)$, choose a <quantifier-free formula> $\psi$ equivalent to it modulo $T$. A <structure embedding> preserves and reflects <atomic formulas>; induction through the Boolean connectives therefore preserves every <quantifier-free formula>. Thus
$$
\mathcal M\models\varphi(\bar a)
\iff\mathcal M\models\psi(\bar a)
\iff\mathcal N\models\psi(j\bar a)
\iff\mathcal N\models\varphi(j\bar a).
$$
The embedding is elementary. \b[Every theory with <quantifier elimination> is <model-complete>.]
Back to article page
