Let $\mathfrak A,\mathfrak A^*$ be $\mathcal L$-structures and $\mathfrak A \preceq \mathfrak A^*$. That implies forall n-ary formula $\varphi(\bar{v})$ in $\mathcal L$ and $\bar{a} \in \mathfrak A^n$ $\models_{\mathfrak A}\varphi[\bar{a}] \iff \models_{\mathfrak A^*}\varphi[\bar{a}]$
Therefore forall $\mathcal L$-sentence $\phi$
$\models_{\mathfrak A}\phi \iff \models_{\mathfrak A^*}\phi$
,which implies $\mathfrak A \equiv \mathfrak A^*$
But I haven't found this result in textbook, so I'm not sure.