Are $\Pi_2$ consequences of universal theories witnessed by finitely many terms?

Here is a question in model theory I need help. Suppose $T$ be a set of universal sentences and $T\models\forall x\exists y\:P(x,y)$. Prove that there exists terms $t_1(x),\cdots,t_n(x)$ such that $$ T\models\forall x \bigvee^n_{k=1}P(x,t_k) $$ Intuitively, it is obvious true but I need a rigorous proof. Does $T$ universal mean that there is a predicate $\phi=\forall x\forall y \:Q(x,y)$ that $\phi\implies \forall x\exists y\:P(x,y)$? Then it follows by the compactness theorem.

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