What happens if the segment between two vertices of a polyhedron is the intersection of the polyhedron with a half-space?

Question

I think that given any two vertices, say $x$ and $y$, of a convex polyhedron, if the line segment between $x$ and $y$ is the intersection of the polyhedron with a half-space, then $xy$ forms an edge. How do I formally prove this fact? I think that somehow I've to use convexity, but I'm not sure how. Please help.