  We consider the mixed local-nonlocal semi-linear elliptic equations driven by
the superposition of Brownian and L\'evy processes \begin{equation*} \left\{
\begin{array}{ll}
  - \Delta u + (-\Delta)^s u = g(x,u) & \hbox{in $\Omega$,}
  u=0 & \hbox{in $\mathbb{R}^n\backslash\Omega$.} \\ \end{array} \right.
\end{equation*} Under mild assumptions on the nonlinear term $g$, we show the
$L^\infty$ boundedness of any weak solution (either not changing sign or
sign-changing) by the Moser iteration method. Moreover, when $s\in (0,
\frac{1}{2}]$, we obtain that the solution is unique and actually belongs to
$C^{1,\alpha}(\overline{\Omega})$ for any $\alpha\in (0,1)$.
