phi_x

$\forall x_1\forall x_2\dots\forall x_{n-1},\exists x_{n}{{\neg{x_{n}=x_1}}\land\neg{x_n=x_2}\land\dot s,\land{\neg{x_n=x_{n-1}}\$
If $\phi_n$ is $\forall x_1,\dots,x_{n-1}.\,x_n\ne x_1\land\dots\land x_n\ne x_{n-1}$, then $\phi_n$ is not a sentence since $x_n$ is a free variable.
A remark concerning notation. Subscripts consisting of more than one symbol must be enclosed in parentheses or braces. Otherwise, it is not clear whether X_n-1 means $X_n-1$ or $X_{n-1}$. You can use /\, \/ and ~ to denote conjunction, disjunction and negation, respectively, in ASCII. It is also not too hard to write LaTeX code for these formulas. The formula above is produced by $$\forall x_1,\dots,x_{n-1}.x_n\ne x_1\land\dots\land x_n\ne x_{n-1}$$. Also, \lor produces $\lor$, \neg produces $\neg$ and \exists gives $\exists$.