That will support gaming for a long time...look at it this way. Right now people run games efficiently with Core 2 Duos.... a Core i7 is an enthusiast CPU. It will be great. Maybe someday you may get another 4870 and CF them, but that setup will be fine. You can also increase the RAM as you need to.
Maybe two years on the CPU if you don't overclock it, 3 if you do. For the GPU maybe 1.5-2 years depending on your resolution and what games you're going to play on it, and of course if you add another 4870 down the line that would certainly help.
Also depends on what you mean by "obsolete." To keep running games at full throttle, you'll have to upgrade, but that machine can continue playing games at high settings for quite some time. I had an entry level computer last 3 years before I decided it was past it's time. A high end enthusiast machine could probably stretch longer.
Obsolete is a silly word. There will be better parts for every aspect of your rig in a month.
How long will you get to game out of it? Well that is up to you. My brother games on an old althon x2, with a radeon 2600.. plays WOW well enough for him.. I go nuts not being able to play at max settings at 1900*1200 at all times. He upgrades every 3 years.. I tend to always be upgrading something..
That rig will last you a long time though, no doubt. How long is entirely up to you.. I bet it will still play games at medium settings for 4 years... May not be able to keep up on high settings as soon as dx11 comes out.. who knows.