Finite subaxiomatization from an equivalent finite theory
= Finite subaxiomatization from an equivalent finite theory
Suppose theories $S$ and finite $T$ have the same models. Let $\tau$ be the conjunction of $T$. Since $S\models\tau$, compactness supplies a finite $S_0\subseteq S$ with $S_0\models\tau$. Conversely $\tau\models S$, so $S_0$ and $S$ have the same models. Thus an axiomatization equivalent to a finite theory contains a finite equivalent subaxiomatization.