Commuting cohomologies for bounded double complex

Question

For a bounded double complex $D$ with differentials $d_1,d_2$ we know that $H(H(D,d_1),d_2)=H(H(D,d_2),d_1)$.

Does anyone know of a reference for the proof without using spectral sequences?

Thanks!

EDIT: I should add both spectral sequences degenerate on page 2.

Answer 1

This isn’t true. For example, take the double complex with the only two nonzero rows $$\require{AMScd} \begin{CD} \cdots@>>>0@>>>0@>>>\mathbb{Z}@>>>\mathbb{Z}@>>>0@>>>\cdots\\ @.@VVV@VVV@VVV@VVV@VVV\\ \cdots@>>>0@>>>\mathbb{Z}@>>>\mathbb{Z}@>>>0@>>>0@>>>\cdots \end{CD}$$ where the maps between copies of $\mathbb{Z}$ are all isomorphisms.