The original proof of Borel determinacy by Donald Martin (1975, Annals of Mathematics) used a priority argument; I haven't read that paper, but Martin cited the complexity of the original proof as a motivation in the paper where he published his second, "purely inductive", proof (1982, 1985, Proc. Sympos. Pure Math. 42). The second proof eliminated the priority argument.
Note: the second paper was published in 1985, but was presented in 1982.