Lemma 11.3. Assuming that:

• $T$ is an $L$-theory

• for any quantifier-free formula $\phi (x_1,\dots ,x_n,y)$ there is a quantifier-free formula $\psi (x_1,\dots ,x_n)$ such that

  $$T\models \forall \overline{x}(\exists y\phi (\overline{x},y)\leftrightarrow \psi (\overline{x})).$$