BC = Backwards Compatible. The 80-Gig model PS3 is the one that came with the Metal Gear Solid bundle, and was the last PS3 model released in the U.S. to feature any Backwards Compatibility. It also had four USB ports, as opposed to the current two, and all of the memory card slots on the front. Although its BC is software-based, and thus not as complete as the hardware-supported earlier models, it is still considered to be one of the best versions of the PS3 available. (the previous 60-Gig models have been known to have issues with their lasers, a problem that had been cleared up by the time the 80-Gig came around)
So yes, $175 would probably be a great price for a fully-functional 80-Gig PS3. In fact, I would strongly recommend that you NOT trade it in to acquire a PS3 Slim. Although the PS3 Slim's form factor is quite attractive, you would technically have a more full-featured and capable package with the 80-Gig model.