Proof of Robinson's test

Question

I have been working with Tent and Ziegler's Model Theory.

I am on the Quantifier elimination chapter, and there they mention Robinson's test. It says that, for an $L$-theory $T$ three statements are equivalent:

1. $T$ is model complete
2. Given two models $\mathfrak{A}_1$ and $\mathfrak{A}_2$ of $T$ such that $\mathfrak{A}_1\subset \mathfrak{A}_2$, and given any existential sentence $\varphi$, then $$ \mathfrak{A}_2\models\varphi \implies \mathfrak{A}_1\models \varphi.$$
3. Each formula is, modulo $T$, equivalent to a universal formula.

I understand the proofs of $3\implies 1$ and $1\implies 2$, but I can't do $2\implies 3$ (and their comments in the book didn't help).

How can you proof that implication?

Answer 1

Hint: Assume 2. Let $\phi$ be any sentence. Consider the set $\Sigma = \{\psi \colon \psi \text { is universal and } T \models \phi \to \psi\}$. Show that $T \cup \Sigma \cup \{\lnot \phi\}$ is not consistent. Deduce that for some finite subset $\Delta \subset \Sigma$ we have $T \models \Delta \to \phi$. Take as your universal formula the conjunction of formulas in $\Delta$.