Existence of Disjunctive Normal Form of Statement

Theorem
Any statement form can be expressed in disjunctive normal form (DNF).

Proof
A simple statement is already trivially in disjunctive normal form (DNF).

So we consider the general compound statement $$S$$.

First we convert to negation normal form (NNF).

This is always possible, by Negation Normal Form of Any Statement.

Now $$S$$ will be of the form:
 * $$P_1 \lor P_2 \lor \cdots \lor P_n$$

where $$P_1, P_2, \ldots, P_n$$ are either:
 * Literals;
 * Statements of the form $$\left({Q_1 \land Q_2 \land \ldots \land Q_n}\right)$$

If all the $$Q_1, \ldots, Q_n$$ are literals we have finished.

Otherwise they will be of the form $$Q_j = \left({R_1 \lor R_2 \lor \ldots \lor R_m}\right)$$

If the latter is the case, then use the Rule of Distribution to convert:
 * $$Q_1 \land Q_2 \land \ldots \land \left({R_1 \lor R_2 \lor \ldots \lor R_m}\right) \ldots \land Q_n$$

into:

$$ $$ $$ $$

It can be seen then that each of the
 * $$\left({Q_1 \land Q_2 \land \ldots \land Q_n \land R_k}\right)$$

are terms in the DNF expression required.

If any terms are still not in the correct format, then use the above operation until they are.