The transfer principle holds for definable nonstandard models under countable choice

¡os's theorem for (bounded) D-ultrapowers, D being the ultrafilter introduced by Kanovei and Shelah [Journal of Symbolic Logic, 69(1):159-164, 2004], can be established within Zermelo-Fraenkel set theory plus Countable Choice (ZF+ACw). Thus, the Transfer Principle for both Kanovei and Shelah's definable nonstandard model of the reals and Herzberg's definable nonstandard enlargement of the superstructure over the reals [Mathematical Logic Quarterly, 54(2):167-175; 54(6):666- 667, 2008] can be shown in ZF+ACw. This establishes a conjecture by Mikhail Katz [personal communication].