$\left ( \forall x \right )\left [ P_{x}\Leftrightarrow \left ( \forall y \right ) \left [ Q_{xy}\Leftrightarrow \sim Q_{yy} \right ] \right]\Rightarrow \left ( \forall x \right )\left [ \sim P_{x} \right ] $
$P$ is a first order formula with $x$ sa its only free variable.
$Q$ is a first order formula with $x, y$ as its free variables.
I want to convert the above formula into its equivalent english sentence, how should I approach?
One way to render that sentence is:
"If for all $x$, $P(x)$ holds exactly when for all $y$, $Q(x,y)$ and $\neg Q(y,y))$ are equivalent, then for all $x$, $\neg P(x)$."