At the bottom of page 58 of Jaakko Hintikka's Principles of Mathematics Revisited he gives a formula in his IF first-order logic
$(\forall x)(\forall z)(\exists y / \forall z)(\exists u / \forall x)(((x=z)\leftrightarrow (y=u)) \& \mathrm{H}(x,y) \& \mathrm{H}(z,u) \& \sim \mathrm{H}(x,u) \& \sim\mathrm{H}(z,y))$
which he says is equivalent to the second-order formula (top of page 59)
$(\exists f)(\forall x)(\forall z)(((x\neq z) \supset (f(x)\neq f(z))) \& \mathrm{H}(x,f(x)) \& \mathrm{H}(z,f(z)) \& \sim \mathrm{H}(x,f(z))\&\sim \mathrm{H}(z,f(x)))$
He says if $\mathrm{H}(x,y)$ is read "$y$ is a hobby of $x$'s" then these say something like "everybody has a unique hobby''. (He also says that these cannot be reduced to any expression in ordinary first-order logic.)
Is there a mistake in those formulas? When $x=z$ then they both require $\mathrm{H}(x,y)$ and $\sim \mathrm{H}(x,y)$. If there is a mistake, what could he have meant? If there isn't a mistake, what's my mistake?