law of implication

I'm trying to follow the solution of an exercise that asks to use rules of inference to show that something is true but I don't know how to go from step 2 to step 3:

Step 1 $\quad \forall x((\lnot P(x) \land Q(x)) \rightarrow R(x))$ (given)

Step 2 (Universal Instantiation) $\quad (\lnot P(a) \land Q(a)) \rightarrow R(a)$

Step 3 $\quad\lnot R(a) \lor \lnot(\lnot P(a) \land Q(a))$ (law of implication)

Any details on how the law of implication was used on step 3 would be appreciated. Thanks!

1 Answer

First, the equivalence of an implication with its contrapositive is used.

$$ (\lnot P(a) \land Q(a)) \rightarrow R(a) \equiv \lnot R(a) \rightarrow \lnot(\lnot P(a) \land Q(a))$$

Then we can use the law of implication: $$\lnot R(a) \rightarrow \lnot(\lnot P(a) \land Q(a))\equiv \lnot \lnot R(a) \lor \lnot(\lnot P(a) \land Q(a)) \\ \\ \equiv R(a) \lor \lnot(\lnot P(a) \land Q(a))$$

So the third line is in error: we need $R(a)$ and not $\lnot R(a)$ to obtain the disjunction in $(3)$.