You actually don't need Vista at all. Here's what you can do:
1) Boot from the Windows 7 CD and install it without the product key (skip it during setup).
2) Run the Windows 7 installer again to do a clean install, but this time from your new Windows 7 install (important that you don't boot from it, also use the product key this time)
3) Delete the windows.old folder in the root directory of your drive (contains files from the old Win7 install)