not just a 7870... i got a tiny little 7770, and am playing games in 1080p on ultra or high.
but i agree with fishwithadeagle, if you want performance at the expected price range for that card, a reference 7870 with some aftermarket cooler like an accelero extreme 2/3, and unlocked voltages will probably give you a lot more bang for the buck at a lot more quiet levels then any "elite" version of the gpu out there. and if you don't care about a 20% boost in performance the stock version is good too.
the 7870-660-660ti-760-7950 are all as much or more gpu then you need if you're gaming at 1080p