GFI wire on PSU Case mod (3 wire)

I'm installing a computer inside an amplifier case and well the psu doesn't have much room, So A. I have to remove it from the case, B I have to mount is inside the plastic front.
So the question is the GFI wire (3rd-wire) that is on one of the four corners of the PSU board and from there to the psu case, should I take another wire from there and also connect it to the metal of my case?
I mean the whole point is to prevent the case during an accidental short from electrocuting peeps. So should I put do that in-order to trigger the fuse from the mains if the power accidentally lives the case?
terramir