Jared - thanks for catching that typo. I fixed it.
Anindya and Owen - thanks for catching that much-worse-than-a-mere-typo. I was rushing in writing this post (as I so often am) and I screwed up. I usually use the double complement way of thinking about \\(f_!\\) but I didn't want to talk about that because I'm avoiding negation so far. I knew the double complement trick would turn a "for exists" into a "for all" since
$$ \forall = \neg \exists \neg $$
But I just randomly grabbed a way to cross out a "there exists" and write a "for all".
I'll fix my post later today - but so people know what the mistake was after it's gone, I've added a clearer description of it to Anindya's original comment.