In a proof that a certain theory $T$ in conservative over $\textsf{ZFC}$, the author makes the following step:
(Here $\Delta$ is a finite set of formulas such that $\textsf{ZFC}\vdash\Delta$.)
Suppose that $\Delta\not\vdash\psi$. Then, by the Levy-Montague reflection theorem, there is some ordinal $\lambda$ such that $(V_{\lambda},\in)\vDash\neg\psi$.
Why does this follow? I understand that we can reflect the set $\Delta\cup\{\neg\psi\}$ at some level of the hierarchy. But can we really get a model $(V_{\lambda},\in)$ satisfying $\Delta\cup\{\neg\psi\}$ just from the assumption that there exists some model of $\Delta\cup\{\neg\psi\}$?