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I'v not had that problem with a corolla, but I did have the sme problem with a Dodge Neon. If it shuts off suddenly, then the problem cant be the battery. On all cars, the battery is used to only start the car, and then the alternator keeps it going. And I'm quite sure it cant be your alternator, because when that goes bad (or begins to go bad), the car would use the battery, slowly draining it until the car stalls. At least with my Neon, there where two red cables coming off of the positive side of the battery, a large one, and a smaller one. The smaller one needed to be connected to the battery (and it was also connected to the alternator), but it was corroded, and sometimes not getting a good connection (thus not getting power from the battery OR the altinator). Once the car it started, even disconnecting the battery shouldn't cause the car to stall, because the alternator is then used to power the car.
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