Computer crashed. If I back up my iPhone on my new computer, would I be able to sync with new computer without losing everything?

The computer I originally synchronized my iPhone with crashed, so I can't add music with my other computer without erasing what I already have. If I made a back up file on the new computer and put the file back on my iPhone, would I be able to add stuff from the same computer without losing anything? Would using the back-up file on that computer count as my iPhone now being synchronized with it?