Bahmanyar – the famous disciple of Avicenna – has a short, little known treatise entitled On the levels of existents (ed. S. Poper, Leipzig: 1851). The work is rather technical, assuming a lot on the part of its reader. It is devoted to issues involving four kinds of immaterial beings (i.e., God, the separate intellects, celestial, and human souls). In chapter 3 of the work, in the course of attempting to establish the existence of immaterial entities other than God, Bahmanyar offers an extremely concise version of an argument Avicenna sets up in The Healing; Ilahiyyat IV.3. I call it ‘proof from the hylomorphic constitution of bodies’.
The argument goes as follows (ibid., p. 22):
The body is composed of matter and form. And the form is not independent in its existence from the matter, nor the matter from the form. Hence, there inevitably exists a third [entity] that is not a body.
Let me ever so slightly unpack the thought in the matn. Take it as established that bodies are hylomorphic composites. If so, then the reason a body exists now and continues to exist for the duration of its temporal career must be because its hylomorphic parts do so.
But then what about its hylomorphic parts? Corporeal form only exists alongside matter, and vice versa. If then they are existentially interdependent, in the sense that neither one can exist without the other, what explains their being?
Their interdependence precludes either of them from doing the job. (This is implicit in the argument). Thus, it must be some third entity apart from both. And, moreover, this causal entity can’t itself be a body; for recall that the argument concerns a body’s existence now, so that all bodies as such are subject to its force. (This too is implicit in the argument). And thus, there exists a third immaterial entity that is the cause of both.