How to sync my iPhone to a new computer without losing data?

I want to download iOS7, but I obviously want to back up my phone before doing so.
My phone is registered with iTunes on my laptop (which by the way, is broken and won't turn on) so I thought about plugging my iPhone into my brother's computer to back it up there.
But he has his iPad synced to that computer, so is there any way I can back up my phone to his computer without losing my data?

You can't.

When you plug your phone in to his computer, iTunes should ask if you would like to "erase & sync" or "transfer purchases" try transferring your purchases onto his computer (this will put all of your music, etc.into his iTunes) and you should be able to sync your phone from his computer.

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