Syncing my Iphone 5 onto a different computer's ITunes?

So I've been using my dad's computer to sync my iphone 5 and buy music from the library and stuff, but recently his computer broke and isn't working, so I have been wanted to just use my mom's laptop because I do have some music on there from previous Ipod's and things i'd like to have. But I had a question. Since you can't sync an iphone to two computers, if I back up my Iphone to my mom's computer, then do the "Erase and Sync" and restore it to that same back up, will my iphone then be synced on my mom's computer, and will I be able to get music from her computer, all while still keeping everything I backed up? I'm not sure if this makes any sense but I just really want to be able to use my mom's computer now for getting music and syncing stuff onto my iphone, but I also don't want to have to erase all the data that was already on it! My dad's computer won't probably be fixed for a while so I was just going to use hers from then on if it did work.

If you want to be sure you can download iExplorer or iPhone explorer. You can download all data through it even when you don't have iPhone synced with this computer.

Its better if you download music from www.albumatom.com
you can get any album for download in mp3 for free
and then you can easily put it on your iphone/ipod/ipad
you can use SharePod software to sync music.itunes is horrible

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