How to solve this logic exercise?

Question

It turns out that there is a determined exercise that I was looking at, but I do not understand how it reaches the following conclusion:

1) $P \Rightarrow \neg [ (\neg P \Rightarrow Q) \land \neg(\neg P \land R) ]$

2) $P \Rightarrow \neg \big[(\neg(\neg P) \lor Q) \land (\neg (\neg P) \lor \neg R)\big]$

3) $P \Rightarrow \neg \big[(P \lor Q) \land (P \lor \neg R)\big]$

4) $P \Rightarrow \neg [ P \lor (Q \land \neg R) ]$

I do not understand how to get from step $3$ to $4$. It justifies saying that distributes with the connectors $\lor$ and $\land$.

Thanks.

Answer 1

To get from step 3) to step 4) in your exercise, change the expression inside brackets from

$(P \lor Q) \land (P \lor \lnot R)$ to $P \lor (Q \land \lnot R),$ using the distributive property.

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Note: there is also a distributive property that $(p\land (q\lor r))\vdash ((p\land q)\lor (p\land r))$