Will a Straight Talk AT&T SIM card work with a new AT&T phone?

[AndroidTerry]

Several months ago, I bought a used unlocked AT&T branded Samsung Galaxy S3 mini off Ebay. For service, I bought a Straight Talk starter kit from Walmart and inserted the AT&T SIM card. Unfortunately, the screen is going out on it so I was thinking about buying a new S4 mini from a local AT&T store. However, being new, this will mean that the phone will be locked. Basically, what I'd like to know is if this same SIM card will work in the new AT&T phone even though it's a Straight Talk AT&T SIM card and not a "direct" AT&T sim card?

 

[AndroidTerry]

Basically what I need to know is if a Straight Talk AT&T SIM card will work with a new, locked AT&T phone.