Page "Constructivism (mathematics)" Paragraph 23

However, in certain axiom systems for constructive set theory, the axiom of choice does imply the law of the excluded middle ( in the presence of other axioms ), as shown by the Diaconescu-Goodman-Myhill theorem.

Some constructive set theories include weaker forms of the axiom of choice, such as the axiom of dependent choice in Myhill's set theory.