I'm having difficulty finding this result in the standard texts.



> 
> **Theorem.** Let $T$ be a theory in a language $\mathcal{L}$. TFAE:
> 
> 
> 1) $T$ has quantifier elimination,
> 
> 
> 2) Whenever $M, N$ are $\aleph\_1$-saturated models of $T$, $A \subset
>  M$, $B \subset N$ are countable nonempty substructures and $f : A
>  \rightarrow B$ an $\mathcal{L}$-isomorphism, then for any $a \in M$
>  there exists an extension $f' \supset f$ with $a$ in the domain of
>  $f'$ and $f'$ still an $\mathcal{L}$-isomorphism. In addition, for any
>  $b \in N$ there exists an extension $f'' \supset f$ with $b$ in the
>  range of $f''$ and $f''$ still an $\mathcal{L}$-isomorphism.
> 
> 
> 


I'm obtaining this result from these notes by Chatzidakis (see 2.27): <http://www.math.ens.fr/~zchatzid/papiers/CUP-MT.pdf>. Any suggestions are appreciated.



Reference request: quantifier elimination test
I'm not sure if this formulation with $\omega\_1$-saturated models is in any of the "main standard" texts. But there is a complete treatment in these [course notes](https://www3.nd.edu/~apillay/pdf/lecturenotes_modeltheory.pdf) by Pillay (see Proposition 2.29).


If you're looking for something in a standard published text, Corollary 3.1.6 of *Model Theory: An Introduction* (Marker) is along the same lines. It says:



> 
> Let $T$ be an $L$-theory. Suppose that for all quantifier-free formulas $\phi(\bar{v},w)$, if $M,N\models T$, $A$ is a common substructure of $M$ and $N$, $\bar{a}\in A$, and there is $b\in M$ such that $M\models\phi(\bar{a},b)$, then there is $c\in N$ such that $N\models \phi(\bar{a},c)$. Then $T$ has quantifier elimination. 
> 
> 
> 


It is not too difficult to deduce the theorem you quoted from this. 


