In Box and Tiao (1973) page# 156, authors write that if the distributions of two random variables are identical except location, then the distribution of the differences would certainly be symmetric. In other words, if two random variables are identical except in their mean, then the difference of the two random variables would be symmetric.
But authors have not provided any proof for this claim. Maybe because it is supposed to be obvious. But I am not able to understand why this statement is true? It will be helpful if someone can show a proof for this claim or an intuitive explanation