How do I prove this simple result for the face structure of convex sets?

Question

I have a convex set $P$ with faces $f_1, f_2$ such that all extremal points of the convex set belong to either $f_1$ or $f_2$ (the faces are disjoint and cover $P$). How can I prove that if every extremal point of $f_1$ shares an face edge with every extremal point of $f_2$ then $\mathsf{dim}(f_1) + \mathsf{dim}(f_2) < \mathsf{dim}(P)$? Examples include any polygon based pyramid in $d=3$, where $f_1$ is the base and $f_2$ is the vertex at the top of the pyramid (and $0+2 < 3$), or the two edges of a tetrahedron that do not share vertices (but for which every vertex of one shares an edge with every vertex of the other), and $1+1 < 3$.