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The 'un-throttling' (dethrottling?) - does that happen automatically if your change the battery yourself - or only if apple do it? I have a iPhone 5S that is struggling...
iPhone 5S doesn’t have the throttling software only iPhone 6 and up.. Replacing the battery may help with performance but, as with the iPhone 6 the hardware is just getting pushed to it’s limits with each new iOS iteration. iOS 12 is supposed to help the older phones perform better so you may get some relief when that officially is released
 
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