Ping-pong and Outer space

We prove that if $\phi,\psi\in Out(F_N)$ are hyperbolic iwips (irreducible with irreducible powers) such that $<\phi,\psi>\le Out(F_N)$ is not virtually cyclic then some high powers of $\phi$ and $\psi$ generate a free subgroup of rank two, all of whose nontrivial elements are again hyperbolic iwips. Being a hyperbolic iwip element of $Out(F_N)$ is strongly analogous to being a pseudo-Anosov element of a mapping class group, so the above result provides analogs of "purely pseudo-Anosov" free subgroups of $Out(F_N)$.