Terminator 3 Question?

Ok, I know this is a a pretty late question, but at the end of Terminator 3, John Connor and that woman are put in the bunker, then they realise there is no CORE to skynet and he gets annoyed. And Judgment Day happens bla bla bla. So... does that mean John Connor goes on to do all the stuff in the first two movies again, if so, does John Connor actually sort out skynet in the end and save the world? Because the last Terminator was a prequel so it didn't actually answer anything.