P. Aczel and S. Feferman [To H. B. Curry: Essays on combinatory logic, lambda calculus and formalism, 67-98 (1980; Zbl 0469.03006)] proposed two new formal systems, as possible foundations for set theory. Both have the abstraction axiom, \(\forall y[y\in \{x| \phi(x)\}\equiv \phi(x)]\), where \(\equiv\) is a weak equivalence relation additional to the classical one. The current paper proposes alternative ”tidier” axiomatizations of the Aczel-Feferman systems, this time using the additional primitive predicate D, for determined (undetermined means neither true nor false). This D was a defined predicate in the Aczel- Feferman system. In proving his systems equivalent to the Aczel-Feferman systems the author uses some results from the reviewer’s paper in Z. Math. Logik Grundlagen Math. 28, 269-276 (1982; Zbl 0496.03034).