$($a$)$ Construct a proof, using only the standard natural deduction rules,

that $|--((AB)harrnot(A^{{?}^{?}}$notB$)).$

$($b$)$ The proof you have just provided shows that $(AB)$ and not$(A^{{?}^{?}}$notB$)$are provably equivalent. That justifies the following derived rules,

which allow you to swap the former for the latter, and vice versa: ${?}^2$

$1|[(AB),1,$not$(A^{{?}^{?}}$notB$)],[vdots,$vdots$],[not(A^{{?}^{?}}$notB$)$ SWAP, $1,(AB)$ SWAP, $1]$

Consider the modified natural deduction system which includes these

SWAP rules in place of the introduction and elimination rules gov$-$

erning the conditional. Show that in this modified system it is possible

to establish our familiar conditional rules as derived rules.