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??