Looking over some of my past work where I planned to convert expressions into 3CNF, I found the following step:
$A \lor \neg B \lor C\lor (\neg D \wedge E)\equiv (A \lor\neg B \lor C \lor\neg D)\wedge(A\lor\neg B\lor C \lor E)$
Can someone elucidate what rule(s) are used to establish that this is true?
Firstly, $\lor$ is associative, so we may suggestively write e.g.:
$$A \lor \neg B \lor C \lor (\neg D \land E) = (A \lor \neg B \lor C) \lor (\neg D \land E)$$
If we then define $X = A \lor \neg B \lor C$, we are left with:
$$X \lor (\neg D \land E) = (X \lor \neg D) \land (X\lor E)$$
which is precisely the statement that $\land$ distributes over $\lor$.
Together these statements yield the desired identity.