 All of the results in set theory which do not use FA go through when one replaces it by AFA.  In particular,  the following topics are unchanged: Russell’s Paradox  and the Separation (Subset) Axioms; the modeling of ordered pairs, relations and functions;  the natural numbers, real numbers, etc.;  well-orderings and the ordinal numbers; transfinite recursion on well-orders and well-founded relations; the Axiom of Choice; problems and results concerning the sizes of infinite sets. The only difference would be in modeling questions for circularly defined objects of various sorts, as we have been discussing them.     