Conditional-disjunction equivalence:
(p $\implies$ q) $\equiv$ ($\lnot$p $\lor$ q)
To show:
($\lnot$q $\implies$ p) $\implies$ (p $\implies$ $\lnot$q) $\equiv$ ($\lnot$p $\lor$ $\lnot$q)
Attempt:
$\lnot$($\lnot$q $\implies$ p) $\lor$ (p $\implies$ $\lnot$q) by conditional-disjuction equivalence
$\equiv$ $\lnot$($\lnot$($\lnot$q) $\lor$ p) $\lor$ ($\lnot$p $\lor$ $\lnot$q) by conditional-disjunction equivalence
$\equiv$ $\lnot$(q $\lor$ p) $\lor$ ($\lnot$p $\lor$ $\lnot$q) by double negation law
$\equiv$ ($\lnot$q $\land$ $\lnot$p) $\lor$ ($\lnot$p $\lor$ $\lnot$q) by DeMorgan law
I am stuck at the last step.