Just a note, you can't fit 4 double slot video cards into an antec 1200, let alone 5. Nor would you be able to fit 5 cards on the motherboard anyway, you'd need an extender for the 5th pci-E slot. That motherboard is kind of pricey for what you plan on doing, I'm not totally sure I understand whether you are building a second computer for gaming or plan on using this motherboard/cpu, but an AMD sempron will bottleneck the crap out of 6970s, so you won't be able to game on it anyway.

The only cases capable of doing 4 cards are ones with 8+ expansion slots. The only case I know capable of doing 5 cards for under $300 is the Rosewill THOR.

EDIT: Also worth noting that you will be hard pressed to find any 5850s these days. Amazon has been out of stock for about a month and there's no real reason to believe they will get any more anytime soon. Your best bet is to buy used cards or else you will have to go skinny with 5830s, another reason why it might be better to go with 2 cheaper motherboards that can utilize PCI-E x1 slots with extenders.

EDIT #2: The antec quattro is an incredible waste of money. You will likely be able to run 4x 5850s on a quality 750Watt power supply (maybe even less but let's play it safe). You can pick up a quality one of those for about half the price.