Logical Arithmetic: Distribution

Question

Hi I'm reading through "How to Prove it" and I was doing one of the practice problems. The question asks you to simplify as far as possible and they give you the following: $$\neg(\neg p \vee q)\vee (p\wedge \neg r) $$ with this, I applied deMorgan's and double negation to get: $$(p \wedge \neg q) \vee (p \wedge \neg r)$$ My question is whether I'm done at this step or can I distribute it.

Answer 1

By distributive laws: $$ (p \wedge \neg q) \vee (p \wedge \neg r)=p\wedge (\neg q \vee \neg r) $$ which may be further modified as: $$ p\wedge (\neg q \vee \neg r)=p\wedge (\neg (q \wedge r)) $$