I'm working through section 4.3. on model theory from Dirk van Dalen's Logic and Structure (fifth ed.) and am struggling with van Dalen's sometimes sloppy way of presenting proofs.
As usual let a structure $\mathfrak{A}$ be elementarily embeddable in a structure $\mathfrak{B}$ ($\mathfrak{A} \prec \mathfrak{B}$), if $\mathfrak{A}$ is isomorphic to some elementary substructure of $\mathfrak{B}$. Furthermore, let $\hat{\mathfrak{A}}$ be the structure resulting from $\mathfrak{A}$ by adding all the members of its domain as constants.
One crucial lemma in van Dalen's treatment of non-standard models is the following:
$\mathfrak{A} \prec \mathfrak{B} \Leftrightarrow \hat{\mathfrak{B}} \models Th(\hat{\mathfrak{A}})$
where $\models$ is the usual first-order satisfaction relation generalized for sets of sentences and $Th(\mathfrak{A})$ the set of sentences true in $\mathfrak{A}$.
Now, my main source of confusion in the proof of the left-to-right direction is that van Dalen assumes without proof that $\hat{\mathfrak{A}} \models \phi(\bar{a_1}, \ldots, \bar{a_n})$ entails $ \mathfrak{A} \models \phi(\bar{a_1}, \ldots, \bar{a_n})$ (let $\vec{a_i}$ abbreviate the sequence $\bar{a_1}, \ldots, \bar{a_n}$) and $\mathfrak{B} \models \phi(\vec{a_i})$ entails $\hat{\mathfrak{B}} \models \phi(\vec{a_i})$. Why is that so?
Concerning the right-to-left direction it's easy to prove that $\mathfrak{A}$ is a substructure of $\mathfrak{B}$ and that $\mathfrak{A} \models \phi(\vec{a_i}) \Rightarrow \mathfrak{B} \models \phi(\vec{a_i})$ for all ${a_i}$ from the domain of $\mathfrak{A}$. But I don't get the converse direction.
Any help would be greatly appreciated.