Affine principal neighbourhoods from ideal-sheaf vanishing

Let $P$ have an affine open neighbourhood $U$, and let $Y=X\setminus U$. Evaluation gives ${\mathcal{I}}_Y\twoheadrightarrow k_P$ with kernel ${\mathcal{I}}_Y\cap {\mathcal{I}}_P$. If first sheaf cohomology of every coherent ideal sheaf vanishes, a global section $f$ of ${\mathcal{I}}_Y$ can be chosen with $f(P)=1$. Then $X_f\subseteq U$ is a principal open subset of an affine variety and is affine.