[FOM] query regarding omega-completeness of ZFC for Th(L(R))

[Rupert McCallum]

In "On the Question of Absolute Undecidability" Peter Koellner says that assuming a proper class of Woodin cardinals, the theory of L(R) is invariant under forcing, and then goes on to say that this can be paraphrased as: assuming a proper class of Woodin cardinals ZFC is Omega-complete for the theory of L(R). This is not obvious to me because one needs to prove that if V_kappa_1 and V_kappa_2 are both models of ZFC, then L(R)^V_kappa_1 and L(R)^V_kappa_2 are elementarily equivalent. Can anyone explain how this is proved?