I have mini RTW with LX booked in C and I need advice on the procedures during the second part of it. I am flying LX016 from ZRH to JFK, having few days there and then UA863 from JFK to SYD (never done before and my only second trip to USA). I understand that there is airplane change over at SFO and that's where I am struggling. I know that if I did it the other way (SYD-JFK) I would have to collect luggage at SFO, clear immigration and customs and hand it back to UA again. Am I correctly thinking that if doing JFK-SYD, luggage goes through without needing to do anything at SFO?