[j2kun]

I think your argument is avoiding the real issues. One problem is that the simplification of setting y = x^y introduces new solutions that are not solutions to the original equation. While any solution of the original must also satisfy y = x^y, the reverse is not true.

Even worse, the OP's algebraic manipulations assume there exists a solution to the equation. It's not logically sound to assume something exists and use that fact to prove that it exists.