I've been doing Logic equivalences questions and I've been taught that a variable $x$ is bound if it is under any quantifier such as $ \exists $ or $ \forall$ however I've come across a question where in the answer it states that a variable is bound even though no quantifier exists to bound it:
$ \forall y [\exists x P(x,y) \Rightarrow \neg S(y)] $
$ \forall y \forall x [P(x,y) \Rightarrow \neg S(y)] $ by equivalence 37: note that $x$ is not free in $S(y)$.
Equivalence 37 in my notes states
Note: If $x$ does not occur free in $B$
$ \forall x (A\Rightarrow B) $ eqv $ \exists x A \Rightarrow B $
$ \exists x (A \Rightarrow B) $ eqv $ \forall x A \Rightarrow B $
Maybe I'm misunderstanding something here? Can someone please explain what it means?