In proof of reversed triangle inequality $|x - y| \geq ||x| - |y||$, which is derived from triangle equality $|x + y| \leq |x| + |y|$ there is a step where triangle inequality is transformed into
$$|(x-y)+ y| = |x| \leq |y| + |x - y|.$$

What is the logic behind that transformation? If I understood that correctly, we put $x = x - y$ in this case, to make $|x|$ term on LHS. This is also permissible to do for inequality, as inequality will remain the same.

Do I understand it correctly, or is there something else I do miss/do not understand?