Consider any finite list $\Phi$ of axioms of ZFC and any other sentence $\phi$. By the Lévy-Montague reflection theorem, there is some rank-initial segment $V_\theta$ of the universe for which all the sentences in $\Phi$ and also $\phi$ are absolute between $V_\theta$ and $V$. Since the sentences of $\Phi$ are part of ZFC, they are true in $V$ and hence also in $V_\theta$. In particular, $V$ looks upon $V_\theta$ as a model of $\Phi$, according to the truth predicate that it can define for this set structure. Therefore, if $V$ thinks that $\Phi\vdash\phi$, then it will think that $V_\theta\models\phi$. Since $\theta$ was chosen so that this sentence is absolute, this implies $\phi$ holds in $V$, as desired. So we've established any instance of the implication.
As you noted in the question, we get this implication only as a scheme, a separate statement for each instance, because we have the reflection theorem also only as a scheme.
Addendum. Let me explain that one can also strengthen the conclusion somewhat, by assuming not that the sentences of $\Phi$ are part of ZFC, but rather merely that they are true. In other words, I claim that ZFC proves every instance of the scheme:
If we take $\psi$ to be the conjunction of the sentences in $\Phi$, this generalizes your scheme. But the same proof works here. By the Lévy-Montague reflection theorem, there is $V_\theta$ for which both $\psi$ and $\phi$ are absolute between $V_\theta$ and $V$. Now, if $\psi\vdash\phi$ and $\psi$ is true (in $V$), then $\psi$ is true in $V_\theta$, and so $\phi$ also is true there, and so $\phi$ is true in $V$, as desired.