Theorem 11.4. Assuming that:

• $T$ an $L$-theory

Then the following are equivalent:

• (i) $T$ has quantifier elimination
• (ii) Let $M,N\models T$, $A\subseteq M$, $A\subseteq N$ (substructures). For any quantifier-free formula $\phi (x_1,\dots ,x_n,y)$ and tuple $\overline{a}\in A$, if $M\models \exists y,\phi (\overline{a},y)$ then $N\models \exists y,\phi (\overline{a},y)$.
• (iii) For any $L$-structure $A$, $T\cup D(A)$ is a complete $L_A$-theory.