Simplicify $((A ⇒ B) ⇒ (B ⇒ A)) ⇒( ¬(A∧B) ⇔ ¬(B∨A))$ I am trying to simplicify the expression:
$((A ⇒ B) ⇒ (B ⇒ A)) ⇒( ¬(A∧B) ⇔ ¬(B∨A))$
and stuck here. The definition of $\land$ is give n by $A\land B$ is $\lnot (A\implies \lnot B)$. 
I tried to remove all the $\land$ and $\lor$, but it seems not work well and become messy.  May I please what is the easiest way to go about something like this? Thanks in advance.
 A: $A \implies B$ is logically equivalent to $\lnot A \lor B$. Therefore:
$\lnot(\lnot(\lnot A\lor B)\lor(\lnot B\lor A))\lor(\lnot\lnot(A\land B)\lor\lnot(B\lor A))\land(\lnot\lnot(B\lor A)\lor\lnot(A\land B)).$
simplifying and drawing up a truth table the expression is reduced to $B\lor\lnot A$.
A: $$((A \Rightarrow B)  \Rightarrow (B  \Rightarrow A))  \Rightarrow( \neg (A\land B)  \Leftrightarrow \neg (B\lor A)) = \text{ (Equivalence)}$$
$$((A \Rightarrow B)  \Rightarrow (B  \Rightarrow A))  \Rightarrow (( \neg (A\land B)  \Rightarrow \neg (B\lor A)) \land (  \neg (B\lor A) \Rightarrow \neg (A\land B))) = \text{ (Implication)}$$
$$\neg (\neg (\neg A \lor B)  \lor (\neg B  \lor A))  \lor ( ( \neg \neg (A\land B)  \lor \neg (B\lor A)) \land  (  \neg \neg (B\lor A) \lor \neg (A\land B))) = \text{ (DeMorgan)}$$
$$(\neg \neg (\neg A \lor B)  \land \neg (\neg B  \lor A))  \lor ( ( \neg \neg (A\land B)  \lor (\neg B\land \neg A)) \land  (  \neg \neg (B\lor A) \lor (\neg A \lor \neg B))) = \text{ (DeMorgan)}$$
$$(\neg \neg (\neg A \lor B)  \land (\neg \neg B  \lor \neg A))  \lor ( ( \neg \neg (A\land B)  \lor (\neg B\land \neg A)) \land  (  \neg \neg (B\lor A) \lor (\neg A \lor \neg B))) = \text{ (Double Negation)}$$
$$((\neg A \lor B)  \land (B \lor \neg A))  \lor ( ( (A\land B)  \lor (\neg B\land \neg A)) \land  ( (B\lor A) \lor (\neg A \lor \neg B))) = \text{ (Commutation, Association)}$$
$$((\neg A \lor B)  \land (\neg A  \lor B))  \lor ( ( (A\land B)  \lor (\neg A\land \neg B)) \land  ( A \lor \neg A \lor B \lor \neg B)) = \text{ (Complement)}$$
$$((\neg A \lor B)  \land (\neg A  \lor B))   \lor ( ( (A\land B)  \lor (\neg A\land \neg B)) \land  ( \top \lor \top)) = \text{ (Idempotence)}$$
$$(\neg A \lor B)   \lor ( ( (A\land B)  \lor (\neg A\land \neg B)) \land \top) = \text{ (Identity)}$$
$$(\neg A \lor B)   \lor   (A\land B)  \lor (\neg A\land \neg B) = \text{ (Adjacency)}$$
$$B \lor (\neg A\land \neg B) = \text{ (Reduction)}$$
$$B \lor \neg A$$
