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

$\sum_{n,m=0}^\infty\frac{(-1)^{n+m}}{(n+1)(n+m+1)}=\sum_{n,m=0}^\infty\frac{(-1)^{m}}{(n+1)(n+m+2)}$

I would appreciate any idea on how to prove it.