The Gitik-Shelah theorem is also perhaps an example, first proved with forcing by its discoverers, and then without by Anastasis Kamburelis and David Fremlin independently:

Moti Gitik, Saharon Shelah, Forcing with ideals and simple forcing notions, Israel J. Math., 68 (1989), 129-160.

And the same authors have more in:

More on simple forcing notions and forcing with ideals, APAL, 59 (1993), 219-238.