Is the following equivalence true:

|x+y|=|x|+|y| \Longleftrightarrow (x+y\geq 0\Longrightarrow x+y =|x| +|y|)\wedge(x+y<0\Longrightarrow x+y=-(|x|+|y|)
because i can only prove that:

|x+y|=|x|+|y| \Longrightarrow (x+y\geq 0\Longrightarrow x+y =|x| +|y|)\vee(x+y<0\Longrightarrow x+y=-(|x|+|y|)