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Unless they've recently changed their policy, they don't actually give you a new battery. Instead, they capture your old iPhone and give you a service replacement one. I don't know how they would handle "unlocking" your service replacement one, since it'll have a different IMEI number that the original phone that you're leaving with them.
 
Hmm not sure they'd be able to tell if you use an OEM battery? Don't know.

Sure someone else will!
 
If it's under a warranty, then they're almost certainly not going to offer you the battery replacement.

The program is designed for people who have an out-of-warranty device with a battery problem.

If it doesn't meet the criteria for replacement, there's no reason for Apple to offer you a replacement!
 
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