You are going to make a downgrade by switching from HD 4870 to HD 5670. HD 5670 is level below your current card.it is fine for gaming at 1440x900 but cannot handle games on 1920x1080. What is your rest of the system?
Radeon HD 4870 minimum requires 500(or higher) watts of power supply with TWO dedicated 6 pin PCI-E connectors not just one. You also need a quality power supply from brands like corsair,antec,silverstone,seasonic. If your current power supply is not sufficient then that can be a problem.
Also make sure you are having the latest drivers for your card. Though HD 4870 is old card but it is still capable of handling games on 1920x1080 resolution. If you really want an good upgrade then you need to go with something like Radeon HD 6870 or Geforce GTX 560 Ti.
No, you would need an ATI 5870 at least if you wanted to play at 1920x1080, however that is if you are planning on playing the game at high to highest settings. Otherwise you would need an ATI 5770 at least. (also look at 6000 series from ATI they are a little bit faster for about the same price).
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