Let A be the statement form $\displaystyle (\neg p \to (q \leftrightarrow r))$. Find a statement form B in "relaxed" disjunctive normal form which is logically equivalent to A. Then Show that $\displaystyle A \approx B$ (i.e. derive B from A using logical laws).

Attempt:

Here's my truth table, I've marked with a * the rows where A is true:

So the disjunctive normal form would be:

$\displaystyle B= ((((((\neg p \wedge (\neg q \wedge \neg r)) \vee (\neg p \wedge q \wedger))) \vee (p \wedge (\neg p \wedge \neg r))) \vee (p \wedge (\neg q \wedge r))) \vee (p \wedge(q \wedge \neg r))) \vee(p \wedge(q \wedge r)))$

Is this correct? ("Relaxed" DNF means we must have disjunctions of conjunctions in which eachvstatement variable occurs exactly once).

Now, we must derive B from A using logical laws

$\displaystyle A=(\neg p \to (q \to r))$

$\displaystyle \iff (\neg p \to ((q \wedge r)\vee(\neg q \wedge \neg r)))$ (Equivalence law)

$\displaystyle \iff (\neg \neg p \vee ((q \wedge r) \vee (\neg q \wedge \neg r)))$ (Implication law)

$\displaystyle \iff (p \vee ((q \wedge r) \vee (\neg q \wedge \neg r)))$ (Double negation)

$\displaystyle \iff (p \wedge ((q \wedge r ) \vee \neg (q \vee r)))$ (De Morgan's law)

$\displaystyle \iff ((p \vee (q \wedge r)) \vee (p \vee \neg (q \vee r)))$ (Distributive law)

This is how far I've got. Could anyone please show me how to complete this?

P.S. Here are the list of all laws http://img833.imageshack.us/img833/9242/laws.jpg