Upon revisiting chapter 1 of Robert S. Wolf's "A tour though mathematical logic" I sumbled upon the following Proposition on page 13 :
Suppose that $P$ is a Boolean combination of statements $Q_1,Q_2, \ldots , Q_n$. Then there is a statement that is propositionally equivalent to $P$ and is in disjunctive normal form with respect to $Q_1,Q_2, \ldots, Q_n$, meaning a disjunction of conjunctions of the $Q_i$'s and their negations
This is accompanied by the following "proof":
Essentially, the disjunctive normal form of a statement comes directly from its truth table. Each row of the truth table with output T indicates one of the conjunctions whose disjunction must be taken.
I am a weary of the informality of the proof , and I went searching for a more formal proof.
Most of the potential proofs that I found seem to rely on a through discussion of First order language. Which is discussed later on in the chapter.
So is a more formal proof possible without definition and a discussion of first order logic