It's not important in this question because you clearly stated what you mean, but in general it seems to me that it's a really bad idea to use the notation “BISH” to refer to constructive mathematics without axiom of countable choice, given that “BISH” is normally used (starting from Bridges & Richman, Varieties of Constructive Mathematics) to refer to Bishop's framework, which implies countable choice. So I would recommend finding some other notation, perhaps adding some kind of decoration around “BISH” to mean “without Choice”.
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