In these lecture slides from Princeton University I found the following definition of a vertex of a (convex) polyhedron (p. 11).
A point $x\in\mathbb{R}^n$ is a vertex of a polyhedron $P$ if
1) $x\in P$
2) $\exists n$ linearly independent constraints that are tight at $x$
The definition I am familiar with of a vertex is a face (intersection of $P$ with a supporting hyperplane) of cardinality one.
Can anyone refer me to a proof of the equivalence of these definitions?