By "without AC" I think you mean "with the negation of AC". It turns out that BT+ZF is equivalent to ZF + a weaker version of AC called the Ultrafilter Lemma,

By "without AC" I think you mean "with the negation of AC". It turns out that BT+ZF is equivalent to ZF + a weaker version of AC called the Ultrafilter Lemma, iirc. There is more info in the Wikipedia article about BT. Also, the Hahn-Banach theorem (plus ZF) proves BT.