What does: "for all free variables shown" mean in Set Theory.

Question

I am reading the definition of what it means for a class $A$ to model a formula of the Language of Set Theory. It begin,

Let $A$ be a class and $\phi(x_1,\ldots, x_n)$ be a formula of the Language of Set Theory with all free variables shown.

What does this mean? Does this mean that in the formula $\phi$, the free variables are exactly $x_1\ldots x_n$. Or are the free variables among $x_1, \ldots x_n$, i.e. some may be free and some may not be free?

For example if $\phi$ were the formula $\forall x(x = x_1)$ would we write $\phi(x_1)$ or $\phi(x,x_1)$? The first only shows the free variables while the second shows all variables.

Answer 1

Your first interpretation is correct. In general, when we write "$\varphi(x_1,...,x_n)$" we are indicating that each of the variables $x_1,...,x_n$ is free in $\varphi$. (This is very similar to function notation: the idea is that $\varphi$ can take inputs corresponding to the $x_i$s. The weird bit is that sometimes we don't display all the free variables.)

Answer 2

My guess is that it means that the free variables are precisely $x_1,\dots,x_n$, i.e, each of them is a free variable and all of them appear somewhere in the formula. For example, you wouldn't write $\forall x (x=x_1)$ as $\phi(x_1,x_2)$ but rather $\phi(x_1)$ because $x_2$ doesn't appear in it.