  We present a Coq formalization of the Quantified Reflection Calculus with one
modality, or $\mathsf{QRC}_1$. This is a decidable, strictly positive, and
quantified modal logic previously studied for its applications in proof theory.
The highlights are a deep embedding of $\mathsf{QRC}_1$ in the Coq proof
assistant, a mechanization of the notion of Kripke model with varying domains
and a formalization of the soundness theorem. We focus on the design decisions
inherent to the formalization and the insights that led to new and simplified
proofs.
