The 7970 is on par with the GTX 680, the RX 580 is on par with the GTX 1060.
If you could swing for a gtx 1070 you'd have them all beat, it would be good to update to the latest bios to ensure compatibility. Performance will be limited by your older mobo, ram and cpu.
It will be close, if it doesn't fit you have the option to remove the top HDD cage for a sure fit.
You won't regret it, you'll notice a big performance boost.
What monitor do you have by the way ?