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I would say that it should take around 30 days for the FHA streamline refinance. Since you have all your information together from the last lender you should be able to easily give that information to another lender if you wish to do so. [self promotion deleted by Zillow moderator.
Paying off your mortgage faster via a cash-in refinance is a smart way to build equity while potentially securing a lower rate. If you have an FHA loan, though, you must wait at least 6 months before refinancing with the FHA streamline program.