This paper has the same aim as that of the author's paper reviewed above, but the system of illative combinatory logic on which it is based has an extra primitive constant Q for equality. Also instead of assuming that all individuals are sets it assumes the (incompatible) property that all sets are individuals. Using also some axioms for Q, the author derives the comprehension, pairing, infinity, replacement and union axioms of ZF. The power set axiom can be derived within this framework in one form, but when stated in another reasonable form it is somewhat anomalous. Extensionality can be added to this, at the cost of Q-consistency; choice can be added but grounding fails. In a final section the author shows the relative consistency of this system with one that lacks its axioms for Q. In a later paper [J. Symb. Logic 48, 771-776 (1983; Zbl 0527.03003)] he shows that this system is absolutely consistent in a weak sense.