In the following formula :

I substituted :

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

2) 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 :

expecting to get a tautology.

But unfortunately i did not get a tautology. Why?

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