# Equality in distribution of a sum

When I have 4 random variables, $$A,B,C,D$$ and know that $$A+B \stackrel{d}{=} C + D$$ and $$A \stackrel{d}{=} C$$, does this imply $$B \stackrel{d}{=} D$$?

Going through the definition $$X \stackrel{d}{=} Y$$ when $$P(X \leq z) = P(Y \leq z)$$ for all $$z \in \mathbb R$$ does not seem to lead me pretty far here, but I also cannot think of a counterexample currently.

Can anyone help me with this? Thanks!

Let it be that $$A$$ has non-degenerate symmetric distribution and that $$C=-A$$.
Then $$A\stackrel{d}{=}C$$ and $$A+0=C+2A$$ (so even stronger than $$\stackrel{d}{=}$$) but not $$0\stackrel{d}{=}2A$$