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I just come across the idiom "pull out a date" which looks very common expression but I cannot find it in any dictionary. Please explain it to me.
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Your subject line says "pull out date," not "pull out a date." Seeing the subject line, I thought maybe you meant "pull-by date," which is stamped on a perishable grocery item to indicate the date it should be "pulled," or removed from the shelf.
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I wouldn't say it's a common expression. In fact, the tiny number of Google hits (19) suggests that it is pretty rare. Do you have any more context?

To me it means nothing more than "produce a date", in whatever sense is relevant to the context. It may suggest making up a date at random when one doesn't actually know when something is going to happen, or be ready. For example: "My boss asked me when the project would be finished, so I pulled out a date..."
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 khoff's reply was promoted to an answer.
khoffYour subject line says "pull out date," not "pull out a date."
Well spotted. It is becoming apparent to me lately that I almost never bother to read subject lines carefully!

(Not that I have ever heard of "pull out date"...)