Assume $\cal K$ is a pseudo elementary class. I need to prove that $\cal K$ is an elementary class iff $\cal K$ is closed under elementary substructures. Pseudo elementary class is a class of reducts of all models of some theory. I know there is a way to do this with ultrafilter, but i would like to avoid them in this case and use elementary diagrams. An elementary diagram of $M$ is the set of $L(M)$ formulas that hold in $M_M$.
I want to show that $\cal{ K} = Mod(Th(\cal K))$, so given $L \subseteq L'$, and assume that an $L'$ theory $T'$ can be axiomatized by a set of $L$ sentences plus a set of universal $L'$ sentences ( $Mod(T')=Mod(\Sigma_1 \cup \Sigma_2)$), and then show that the class of all reduces to $L$ f models of $T'$ is $\Delta$ elementary. But I don't know very well how to develop this idea, I seem unable to go past the sketch of my proof, and would very much appreciate any help working this out.