Definition of variables in propositional calculus

Question

Let $\tilde P$ be a first order algebra, and consider the definitions below:

I'm confused about the very last thing: what $y\not\in V(c)$ means. $c$ has a free variable, so what does it mean to say $V(c)$?

(This is from an algebraic introduction to mathematical logic).

Answer 1

It means that :

$w_1=(\forall x)c(x)$

and

$w_2=(\forall y)c(y)$

and : $x \ne y$ and $y$ is not free in $c=c(x)$.

If not, the free occurrence of $y$ in the sub-formula $c(x)$ will be free in $w_1$ while it will be "captured" by the quantifier $(\forall y)$ in $w_2$.