Using the rules of boolean algebra prove (state which rule is used at each step):-

$\displaystyle

\forall x in X[ p(x) \rightarrow \neg ( q(x) \wedge r(x) ) ] \equiv \neg \exists x in X[ p(x) \wedge q(x) \wedge r(x) ]

$

So far I have,

$\displaystyle

\forall x in X[ \neg p(x) \lor \neg (q(x) \wedge r(x)) ] Rewriting \rightarrow.

$

$\displaystyle

\neg \exists x in X[ p(x) \lor \neg (q(x) \wedge r(x) ) ] Extended De Morgan (ii)

$

This is where I hit the brick wall. I've also tried doing the Extended De Morgan rule first but that doesn't seem right to me.

any help appreciated.