We study the time optimal control problem for differential inclusions with a general closed target. We first give a representation of the proximal horizontal subgradients of the minimum time function T and then, together with a known representation of the proximal subgradients, we obtain some relationships between the normal cones to the sublevel set and to the epigraph of T. These relationships allow us to get the propagation of the proximal subdifferential as well as of the proximal horizontal subdifferential of T along optimal trajectories. Finally, we show, under suitable assumptions, that the epigraph of T is phi-convex near the target. This is the first nonlinear phi-convexity result valid in any dimension.