We discuss some highlights of our computer-verified proof of the construction, given a countable transitive set-model $M$ of $\mathit{ZFC}$, of generic extensions satisfying $\mathit{ZFC}+\neg\mathit{CH}$ and $\mathit{ZFC}+\mathit{CH}$. Moreover, let $\mathcal{R}$ be the set of instances of the Axiom of Replacement. We isolated a 21-element subset $Ω\subseteq\mathcal{R}$ and defined $\mathcal{F}:\mathcal{R}\to\mathcal{R}$ such that for every $Φ\subseteq\mathcal{R}$ and $M$-generic $G$, $M\models \mathit{ZC} \cup \mathcal{F}\text{``}Φ\cup Ω$ implies $M[G]\models \mathit{ZC} \cup Φ\cup \{ \neg \mathit{CH} \}$, where $\mathit{ZC}$ is Zermelo set theory with Choice. To achieve this, we worked in the proof assistant Isabelle, basing our development on the Isabelle/ZF library by L. Paulson and others.