Provable soundness of finite fragments of ZFC Fix some reasonable computable enumeration of the axioms of ZFC, and let $ZFC_n$ be the theory consisting of the first n axioms.
Is it the case that, for each natural number n, and each sentence $\phi$, ZFC proves the sentence
$(ZFC_n \vdash \phi ) \rightarrow \phi$?
(Where we formalize provability in ZFC in some reasonable way.)
Please note that this is distinct from asking whether ZFC proves
$\forall n \forall \phi \ : \ (ZFC_n \vdash \phi ) \rightarrow \phi$
(Which ZFC trivially does not, if it is consistent.)
 A: 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:
$$(\psi\vdash\phi)\to(\psi\to\phi).$$
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.
