back up your stuff before proceeding
click on transfer purchases
in itunes you click on
file
device
"transfer purchases from xx's iPhone"
if you want to pull stuff (music, tv shows, movies, videos ) off the iPhone you need to use a different utility, like copy transfer manager.
you can use iphoto to pull photos and videos you shot with the camera off the iPhone
as far as "erasing" the iPhone, when you sync your iPhone to another computer, the only thing that gets really erased on the iPhone is sycned content
if you have anything in videos, music, books, photos, music, podcasts, movies, tv shows, that only exists on your iPhone STOP NOW AND DO NOT CLICK ON SYNC
if all that stuff already exists on your itunes library, then go ahead and tell itunes to erase and sync with a new library
all the important stuff, like iCloud, and email , calendar, notes, settings, apps ,phone book won't get erased
make sure itunes and iPhone are using the same apple id
about backups.
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backups do not back up synced content. if you back up to itunes and your content exists on another computer, it won't be in the backup, and if you restore the iPhone, it will be lost
anything purchased on the iPhone can be redownloaded from apple. apple says you may not be able to download a tv show or an app after we remove it from the store, but that hasn't been my personal experience
do you still have access to the old computer that it was synced too??
you should move that content from that old itunes library and place it to your new itunes library if you want to sync to a different computer
you can only sync to one computer at a time. all syncing does is place content from itunes to your phone.
it won't pull content from the iPhone back into itunes, unless the content was purchased from apple.
as long as all your content on the iPhone exists in itunes at the same time, it is safe to sync
if you ripped cds from your personal music collection and those mp3s only exist on your phone and you do not have them on itunes, when you sync to a new itunes library they are gone forever