No. Backups made on iTunes only affects user-end data. iOS versions are system files and are not user-end data. Say on your VZW 4S on iOS 5.0.1, you back it up, so all your music, photos, videos, apps, etc, are all backed up into one nice little package lets say. Then you plug in your Sprint 4S on iOS 5.1.1, and you "Restore from backup", all of those songs, photos, videos, apps, etc, are all transfered over to your VZW 4S. BUT, your Sprint 4S will STILL REMAIN iOS 5.1.1 because you're only restoring user-end data, and not restoring the iOS itself. They're 2 different things. From what I'm reading, you're trying to transfer your stuff from a VZW 4S to a Sprint 4S. If your VZW 4S is on iOS 6 and your Sprint 4S is on iOS 5, you cannot put the back up on the Sprint 4S because they are on different iOS threshold, one being iOS 6 the other being iOS 5.