Probably not, but it might depend on the phone and if it's unlocked or not. Any phone to be used with AT&T must be a GSM device, either AT&T version or an unlocked one, and should be a quad band phone at least. Verizon used CDMA technology and while some of their phones have a GSM antenna and a SIM card slot, those are meant for international travel and might not work in USA.
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It definitely does depend on a number of factors but for the most part, Verizon 4G LTE phones which are unlocked do work on HSPA+ on AT&T, but not LTE as Verizon uses Band 13 while AT&T uses Band 17 for service on LTE.
MMS might also be a problem but can be fixed by inputting the APN settings.
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