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Answer by Joel David Hamkins for Is Morse-Kelley set theory with Class Choice...

No, because the latter theories has parametrically definable automorphisms that swap two equivalent classes, but the former theory is definably rigid (no need for class choice). If the theories were...

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Is Morse-Kelley set theory with Class Choice bi-interpretable with itself...

Let $\sf MKCC$ stand for Morse-Kelley set theory with Class Choice. And let this theory be precisely $\sf MK$ with a binary primitive symbol $\prec$ added to its language, and the following axioms...

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