Ordinal analysis of non-monotone -definable inductive definitions

Exploiting the fact that -definable non-monotone inductive definitions have the same closure ordinal as arbitrary arithmetically definable monotone inductive definitions, we show that the proof theoretic ordinal of an axiomatization  of -definable non-monotone inductive definitions coincides with the proof theoretic ordinal of the theory  of arithmetically definable monotone inductive definitions.