I chanced upon a seemingly "too good to be true" proof of Demorgan's theorem for boolean algebra, however I'm not quite sure if it's valid.
The principle of duality states that for a boolean algebra, changing all OR signs to AND signs, all 1's to 0's, and vice versa for both statements, gives another boolean algebra.
So let us assume, $\overline A + \overline B = X$
Since all the boolean variables have values of either 1 or 0, complementing them is equivalent to applying the relevant part of the duality principle.
So, $\overline A +\overline B = X$
$\Rightarrow A \times B = \overline X$
$\Rightarrow \overline{AB} = \overline{(\overline X)}$
$\Rightarrow \overline{AB} = X$
And thus, $\overline A + \overline B = \overline {AB}$
Is the above proof valid?