In this paper a theorem will be established which states that a particular axiom of choice is equivalent to complete distributivity of union and intersection. The theorem will be formulated and proved in the system of logic of [4]. In addition to definitions of [4], the following will be used.In terms of these definitions, the theorem can be formulated as follows.The dual of this statement, obtained by interchanging I and U, is also a theorem and has a similar proof.