ABC implies primitive prime divisors in arithmetic dynamics

Let K be a number field, let φ(x) ∈ K(x) be a rational function of degree d > 1, and let α ∈ K be a wandering point such that φ(α) = 0 for all n > 0. We prove that if the abc-conjecture holds for K, then for all but finitely many positive integers n, there is a prime p of K such that vp(φ (α)) > 0 and vp(φ (α)) 0 for all positive integers m < n. Under appropriate ramification hypotheses, we can replace the condition vp(φ (α)) > 0 with the stronger condition vp(φ (α)) = 1. We prove the same result unconditionally for function fields of characteristic 0 when φ is not isotrivial.