Am running a dura ace 7700 9 speed rear derailleur and 7700 9 speed shifters at present. would a 7800 dura ace rear derailleur shift correctly with the 7700 shifters and 9speed cassette . would each click line up correctly or is the parallelogram of the 7800 different from the 7700 thus not moving the same distance for each click as the 7700? Thanks for any imput on this post.

Why would you want to replace one of the nicest looking and functioning rear derailleurs Shimano ever produced?

RD-7800 should work as a replacement for the RD-7700 with no real issues. They use the same actuation ratio.

+1. Will work fine.