Generalizing the Reflection Principle of $ \mathsf{ZFC} $ The Reflection Principle states that for any finite list $ \phi_{1},\ldots,\phi_{n} $ of axioms of $ \mathsf{ZFC} $, we have
$$
\mathsf{ZFC} \vdash
(\exists c) \!
\left(
\text{“$ c $ is countable and transitive”} \land
\left( \bigwedge_{i = 1}^{n} \phi_{i}^{c} \right)
\right),
$$
where $ \phi_{i}^{c} $ denotes the relativization of $ \phi_{i} $ to $ c $ for each $ i \in \{ 1,\ldots,n \} $.
What I would like to know is, if $ S $ is a set of sentences in the first-order language of set-theory, and if $ \mathsf{ZFC} + S $ is consistent, then is the following assertion also true?

For any finite list $ \phi_{1},\ldots,\phi_{n} $ of sentences taken from $ \mathsf{ZFC} + S $, we have
  $$
\mathsf{ZFC} + S \vdash
(\exists c) \!
\left(
\text{“$ c $ is countable and transitive”} \land
\left( \bigwedge_{i = 1}^{n} \phi_{i}^{c} \right)
\right).
$$

Thank you very much for your help.
 A: $\newcommand{\ZF}{\mathrm{ZF}}$I would like to state a version of the Reflection Principle under $\ZF$ that (at least to me) makes the point a little clearer. Informally, it says that the truth value each every formula (not just sentences) gets reflected in many transitive sets.
For this, recall the cumulative hierarchy $V_\alpha$ given by iterated powersets. The precise statement of this version says:

Theorem. ($\ZF$) For each formula $\phi(x_1,\dots,x_n)$, sets $a_1,\dots,a_n$  and each ordinal $\beta$ with $\bar a = a_1,\dots,a_n\in V_\beta$, there exists $\alpha>\beta$ such that
  $$V\models\phi(\bar a) \iff V_\alpha\models \phi(\bar a).$$

Here, as usual, $V$ is the universe of sets. Actually, one can show that for each $\phi$, there is a club of ordinals $\alpha$ for which the conclusion holds. 
In the case of $\phi$ a sentence that holds in $V$ (that is, any consequence of the axioms you put to the left of $\vdash$), you obtain immediately your conclusion$^1$ (without the “countable” adjective). Some care is needed here since intuitively $V_\alpha\models\phi$ is “equivalent” to $\phi^{V_\alpha}$ but the last expression is not a formula. But perhaps you can fill the gaps. 
Finally, using the downward Löwenheim-Skolem-Tarski theorem you may obtain a countable elementary submodel (choose some Skolem functions, take the closure, using that countable union of countable sets is countable verify that it is indeed countable). This is a submodel of a well founded model, hence well founded, and you can collapse it to a transitive model.
I hope this helps; actually, the idea marked with $(^1)$ contains the meat of my answer, since that's how I managed to understand the point of your question.
A: I just thought that I should convert Pedro’s response into a fully direct answer to my question.

Theorem Scheme 7.4 in Nik Weaver’s book Forcing for Mathematicians states the following:

For any sentence $ \phi $ in the first-order language of set theory (henceforth called “LOST”), we have
  $$
\mathsf{ZFC} \vdash
(\exists c)
(“\text{$ c $ is countable and transitive}” \land (\phi \iff \phi^{c})).
$$

This is very similar to the version of the Reflection Principle stated by Pedro.
Let $ S $ be a set of LOST-sentences such that $ \mathsf{ZFC} + S $ is consistent, and let $ (\phi_{i})_{i \in [n]} $ be a finite list of sentences taken from $ \mathsf{ZFC} + S $. Basic logic gives us
$$
\mathsf{ZFC} + S \vdash \bigwedge_{i = 1}^{n} \phi_{i}. \qquad (\clubsuit)
$$
By the quoted result, we have
$$
\mathsf{ZFC} \vdash
(\exists c) \!
\left(
“\text{$ c $ is countable and transitive}” \land
\left(
\bigwedge_{i = 1}^{n} \phi_{i} \iff
\left( \bigwedge_{i = 1}^{n} \phi_{i} \right)^{c}
\right)
\right).
$$
This is equivalent to
$$
\mathsf{ZFC} \vdash
(\exists c) \!
\left(
“\text{$ c $ is countable and transitive}” \land
\left(
\bigwedge_{i = 1}^{n} \phi_{i} \iff \bigwedge_{i = 1}^{n} \phi_{i}^{c}
\right)
\right).
$$
As $ \mathsf{ZFC} + S $ is clearly at least as strong as $ \mathsf{ZFC} $, it follows that
$$
\mathsf{ZFC} + S \vdash
(\exists c) \!
\left(
“\text{$ c $ is countable and transitive}” \land
\left(
\bigwedge_{i = 1}^{n} \phi_{i} \iff \bigwedge_{i = 1}^{n} \phi_{i}^{c}
\right)
\right). \qquad (\spadesuit)
$$
Finally, applying modus ponens to $ (\clubsuit) $ and $ (\spadesuit) $, we obtain
$$
\mathsf{ZFC} + S \vdash
(\exists c) \!
\left(
“\text{$ c $ is countable and transitive}” \land
\left( \bigwedge_{i = 1}^{n} \phi_{i}^{c} \right)
\right).
$$
