Here is a brief proof of the Reflection Theorem given in Kunen's Set Theory. The theorem says briefly that for a given finite set of formulas of set theory, then for any ordinal $\xi$ there is a $\eta> \xi$ such that every formula in the list holds in $V_\eta$ if and only if it holds in $V$.
Proof: Assume WLOG that the list of formulas $\varphi_0,\dots,\varphi_{n-1}$ is subformula-closed. For each existential $\varphi_i(\vec{x})$ (of the form $\exists y\varphi_j(\vec{x},y)$, where $\vec{x}$ is an $r$-tuple). Define $F_i:V\rightarrow ON$ such that If $\varphi_i(\vec{a})$, then $F_i(\vec{a})$ is the least $\zeta$ such that $\exists b\in V_\zeta \,\varphi_j(\vec{a},b)$. If $\neg\varphi_i(\vec{a})$ then $F_i(\vec{a}) = 0$.
Then define $G_i:ON \rightarrow ON$ by: $G_i(\xi) = \sup\{F_i(a_1,\dots,a_r):a_1,\dots,a_r \in V_\xi\}$ when $\varphi_i$ is existential and $0$ otherwise. Finally, let $K(\xi)$ be the larger of $\xi +1$ and $\max \{G_i(\xi):i<n\}$. Fix $\xi$. Let $\zeta_0>\xi$ such that $V_\zeta\neq\emptyset$. Then let $\zeta_{n+1}=K(\zeta_n)$. Then $\xi<\zeta_0<\zeta_1<\cdots.$ Let $\eta = \sup\{\zeta_k:k\in \omega\}$. Then $V_\eta$ satisfies the Tarski-Vaught criterion for classes and finite formulas and we are done.
Where in this proof do we need $\varphi_0,\dots,\varphi_{n-1}$ to be a finite list of formulas? It seems like in the proof we only use $n$ once when defining $K(\xi)$. And If we let the set $\{G_i(\xi):i<n\}$ range over $i<\omega$ we'll still get an ordinal. But this way we are getting a $G_i$ for every formula instead of a finite subset. Why can't we do that and still have the proof work? Is it in the Tarski-Vaught criterion? It seems like we can have a class version of that lemma and still have it range over every formula instead of a subset of the formulas.
The issue is in defining the $F_i$s. Specifically, you don't just need each $F_i$ individually, you need the whole tuple $\mathcal{F}=\langle F_0,F_1,...,F_{n-1}\rangle$.
But this is a fairly complicated object: the truth definition for $\Sigma_k$ formulas gets more and more complicated as $k$ increases (this is unavoidable per Tarski). Now since $n$ is finite there is some $k$ such that each $F_i$ is $\Sigma_k$, and so $\mathcal{F}$ itself can be defined via the $\Sigma_k$ truth predicate. However, if we were to try to form the corresponding sequence for an infinite set of formulas, we would be unable to do this if the quantifier complexities of the elements of that set were unbounded.