How to restore an iPhone 4 (iOS 5.1.1) from a back up of another iPhone 4 (iOS...

...6.1.3)? I am trying to restore an iPhone 4 currently running iOS 5.1.1 (latest of 5, named Aj!'s iPhone) from a backup of another iPhone 4 that runs iOS 6.1.3 (latest of 6, named Swedish Fish). Aj's is an 8gb model whereas Swedish is a 16gb. I made sure to remove enough data from Swedish to fit the capacity of 8gb (<4 as saved in the backup).

I want to restore Aj's from a backup of Swedish, but the backup of Swedish does not show as a list option when I plug in Aj's. Although, it is surely backed up as it appears when viewing the information of Swedish. As shown in the picture, when looking at Aj's phone, there is no "Swedish..." option as when Swedish itself is viewed.

There should be no issue restoring from this backup as it is of a newer iOS, even though not fully updated to iOS 7. As well as the memory requirements and software capabilities being compatible and of the same iPhone 4.

I was thinking I may be able to update it to iOS 7, then restore it back down to Swedish backup, but I was unsure whether this would work.

Any advice on solutions would be appreciated. Thank you.