Minimal sufficient statistic implies any complete statistic is also minimal sufficient

Question

I am reading about data reduction of Casella book. There is a theorem establishing the next:

If a minimal sufficient statistic exists, then any complete statistic is also a minimal sufficient statistic.

A proof is no provided by the author; is there a reference for a formal proof of this fact?

Any kind of help is thanked in advanced.

Answer 1

It is called Bahadur's theorem.

A version of it is proved here as Theorem 12.

I could not find a version where sufficiency of the statistic is not a requirement from the premise.