Exponential system of equations

Question

Find $x,y \in \mathbb{R}$ such that \begin{align*} 2^x+3^y=7, \\ 2^y+3^x=11 \end{align*} Obviously, $(x,y)=(2,1)$ is a solution, but I can't prove that it's the only one. I tried using contradiction arguments, monotony, convexity and $\log$s, but didn't succed.