I was listening to a podcast when I heard IF YOU DON'T BE QUIET, I'LL ...
Is it grammatically correct to use BE with an auxiliary (a helping verb) DO in negatives?
Hopefully it's impossible to say DO YOU BE ... ?
It is possible in imperatives only: Don't be a fool! CB
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Yes, it's fine.
Note that "be" is almost always an auxiliary verb, but here it is a lexical verb.
Your example conveys that you should be quiet (in order to ...).
It bears some resemblance to the imperative, but syntactically it is quite distinct from the imperative construction.