Conditions for unique solution

Conditions for unique solution

It might be too specific to be asked here, but anyway. I have the
following simple equation, $$ x-\Phi(\frac{k-x}{\sigma\sqrt{2}})=0 $$
where $\Phi(.)$ is cdf of the standard normal distribution.The question
is, is this system has a unique solution? I have tried implicit function
theorem but could not figure out how to conclude the answer is
affirmative. Many thanks for every help!
P.S: I am trying to find solution as function of $k$. Say, we have unique
solution $x*$ which is defined as function $b(k)$. So, I am also looking
for properties of $b(k)$ like monotonicity or whatsoever