Can I downgrade an iPhone 4 from iOS 6 to iOS 5.1.1 using iTunes if I have a 5.1.1 restore.ipsw file? I have watched this video on YouTube and was wondering if that works for downgrade. I dont need to sync the phone with any of my files afterwards. I just need to have it as new with 5.1.1 software.
In short: not if you didn't bother about jailbreaking or never heard of TinyUmbrella.
Apple is very strict on installing iOS firmware. Everytime you do a restore, iTunes is verifying the firmware for the connected device with their own server by checking SHSH blobs at Apple's servers. An app like TinyUmbrella allows you backup your SHSH blobs (they are unique for every iOS device) locally or to saurik's server. This however only works the latest iOS version that is available for your iOS device.
Not without jailbreaking (and even then it might not work). Apple generally doesn't go backwards so iTunes needs to be cajoled, tricked or bludgeoned into ignoring the current OS and using an ipsw that it doesn't download and verify with Apple's servers.
Have a look at the site and do some research - you can always ask a narrower question if you decide to try a jailbreak and run into a problem on a certain step.