Prove that if A is hopelessly egocentric, then for every bird x, so is A(x).
Suppose A is hopelessly egocentric. We've already proved that, for any birds x and y, A(x)(y) === A. We also know that A(x) === A, simply because of A's hopeless egocentricity. Since they're both equal to A, A(x)(y) and A(x) are equivalent. And, if A(x)(y) === A(x), A(x) is hopelessly egocentric.
Smullyan's solution here seems unnecessarily complicated. From A's hopeless egocentricity, we already know that A(x) is equal to A, and we already know that A is hopelessly egocentric. Why get y involved?