We say an $\mathcal{L}$-theory $T$ has built in Skolem functions if for all $\mathcal{L}$-formula $\phi(x,\bar{y})$ there is a function symbol $f$ such that $T\models \forall\bar{y}(\exists x\phi(x,\bar{y} )\rightarrow \phi(f(\bar{y}),\bar{y}))$.
$ \textbf{Question}.$ Let a theory $T$ have built in Skolem functions. How can we prove that $T$ has $\forall$-axiomatization?
You just need to combine two facts to prove this:
If a theory $T$ has built-in (also called definable) Skolem functions, then every substructure of a model of $T$ is an elementary substructure.
A theory $T$ has a universal axiomatization if and only if it's class of models is closed under substructure.