I have a system of equations:

$$1/2 + {\rm Erf}(x) - {\rm Erf}(\frac{x+y}{2})=0$$ $$-1/2 + {\rm Erf}(y) - {\rm Erf}(\frac{x+y}{2})=0,$$ Where $x \le y$ and ${\rm Erf}$ is the Error Function.

By ansatz I know that one solution is $(x,y)=(-{\rm Erf}^{-1}(1/2),{\rm Erf}^{-1}(1/2))$, but I can not manage to prove that this is the only solution, even though I strongly suspect it. My question therefore is how to prove uniquness.

One re-arrangement I have tried results in: ${\rm Erf}(\frac{x+y}{2}) = \frac{{\rm Erf}(x)+{\rm Erf}(y)}{2}$, which I am pretty sure is equivalent to $y=-x$ (apparent by implicit plot), but again I can not prove this. Proving that $y=-x$ for all solutions would imply the desired uniqueness.

Any advice on manipulation tactics for the error function would be much appreciated..!

Tagged as probability theory because of the relation to the Normal distribution CDF.