You should have no trouble, including checking luggage (if you need to). I did almost exactly the same thing a couple years ago, even returning on the same LHR-IAD flight, IIRC. The only difference was that I was going to/from DEN instead of PHX. When I checked in at LHR, I just gave them both flight #'s and they took care of everything, including checking my luggage "through" to DEN. (Of course, you still have to pick it up for customs and re-check at IAD, but it's much easier if it's already pre-checeked for the connecting flight.)

The only trick is that you do have to remember to tell the agent about your second flight when you check in, since it's on a separate PNR. I do everything so much on auto-pilot these days that I sometimes forget details like that.