When will the next mortgage payment be due after a refinance?

I am about to apply for a refi on my mortgage. My June payment is due June 1st which is coming up in a couple weeks. If I get approved for the loan, does that push my next payment (first payment to the new back) to July 1st or would it still be due June 1st to the new back? I know when I transferred my car loan to a new bank, it skipped the month payment before starting the billing cycle.

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