Alternate Heegaard genus bounds distance

Suppose M is a compact orientable irreducible 3-manifold with Heegaard splitting surfaces P and Q. Then either Q is isotopic to a possibly stabilized or boundary-stabilized copy of P or the distance d(P)<= 2genus (Q). More generally, if P and Q are bicompressible but weakly incompressible connected closed separating surfaces in M then either P and Q can be well-separated or P and Q are isotopic or d(P)<= 2genus (Q).