Is the following theorem true? If yes, how would you prove it?
Theorem Let $A$ be a commutative ring. Let $A[[x]]$ be the ring of formal power series in one variable. Let $\mathfrak{m}$ be the ideal of $A[[x]]$ generated by $x$. Let $u$ be an invertible element of $A$. Let $f(x) = ux + g(x)$, where $g(x) \in \mathfrak{m}^2$. Then there exists a unique automorphism $\psi$ of $A[[x]]$ fixing every element of $A$ such that $\psi(x) = f$.