Since the question asked "for all B", I let .
But I can't continue.
If it's true for all B, it's certainly true for the particular B's I mentioned in my first post. Your approach might work but it seems a bit of a mess. Note that you're only asked to prove implication in one direction. Also note that the problem statement is a little imprecise in language, I would write "for some alpha" rather than "for any alpha". (Clearly it's an existential rather than universal statement, because the universal statement makes no sense.)