I suspect the following identity is valid, but I can not prove it. I just calculate it numerically.

$\sum_{m=0}^\infty\left[\sum_{n=0}^\infty\frac{(-1)^{n+m}}{(n+1)(n+m+1)}\right]=\sum_{m=0}^\infty\left[\sum_{n=0}^\infty\frac{(-1)^{m}}{(n+1)(n+m+2)}\right]$

I would appreciate any idea on how to prove it.