We study the averaging principle for a family of multiscale stochastic dynamical systems. The fast and slow components of the systems are driven by two independent stable Lévy noises, whose stable indexes may be different. The homogenizing index $r_0$ of slow components has a relation with the stable index $α_1$ of the noise of fast components given by $0<r_0<2-2/{α_1}$. By first studying a nonlocal Poisson equation and then constructing suitable correctors, we obtain that the slow components weakly converge to a Lévy process as the scale parameter goes to zero.