In the following formula :

$\displaystyle |x+y| = |x|+|y|\Longleftrightarrow (x+y\geq 0 \Longrightarrow x+y=|x|+|y|)\wedge(x+y<0\Longrightarrow x+y=-|x|-|y|)$

I substituted :

1)|x+y|=|x|+|y| with a

2) $\displaystyle x+y\geq 0$ with b

3)x+y=|x|+|y| with c

4) x+y<0 with d

5) x+y = -|x|-|y| with e

Then

I tried to write the truth table of :

$\displaystyle a\Longleftrightarrow (b\Longrightarrow c)\wedge (d\Longrightarrow e)$

expecting to get a tautology.

But unfortunately i did not get a tautology. Why?

Does that mean that the above formula in not provable??