8350 is NOT cheaper than a 3470.
You might consider the i5-3350p. It's a got a slower turbo boost than the 3470, but it is often quite a bit cheaper because it lacks the built in video of the other i5's. But since this is for gaming, we can assume you'd have a "real" graphics card anyway.
i5-3350p is $160 on Amazon right now, $40 cheaper than the 3470. Sometimes it's about the same price as 3450/70, though, so it depends on the deals that week. It's only worth it if it's cheap (like now!)
Honestly, all the i5's are all great for gaming. It doesn't really matter that much which one you get.