S-integral preperiodic points for dynamical systems over number fields

Given a rational map $\varphi :{\mathbb{P}}^1\to {\mathbb{P}}^1$ defined over a number field $K$, we prove a finiteness result for $\varphi$-preperiodic points which are $S$-integral with respect to a non-preperiodic point $P$, provided $P$ satisfies a certain local condition at each place. This verifies a special case of a conjecture of S. Ih.