Applying Distributivity law on 2 conjunctive statements seperated by a disjunction

Question

I have been given this statement to convert to CNF:

$$((a \to x) \land (b \to c)) \to (a \to ¬c)$$

and so far I have gotten rid of the implications and applied de Morgan's law after which I have:

$$((a \land \lnot x) \lor (b \land \lnot c)) \lor (\lnot a\lor \lnot c)$$

but I can't figure out how to proceed from this point and how I would apply distributivity laws on this.

Answer 1

Let X, Y, Z be three propositions ( either atomic, or compound,we needn't care):

Distributivity of OR over & means that :

X OR ( Y & Z) is equivalent to (X OR Y) & ( X OR Z).

Any thing , any proposition ( either atomic or compound) can play the role of X, Y or Y.

So, here:

Let (a&~x) play the role of X.

Let b play the role of Y.

Let ~c play the role of Z.

Further transformations may require you to let other propositions play the roles of X, Y and Z.

Maybe, after having applied distributivity once, you'll have to use OR-commutativity in order to be in a position to apply distributivity a second time.