elementary set theory – Propositional expansion of $forall x P(x)$

Wikiproof’s article on universal quantifier states that, considering the finite universe of discourse $x_o, x_1 ldots x_n$, the quantified formula $forall x P(x)$ is equivalent to $P(x_0) land P(x_1) ldots P(x_n)$. I’m having problems to understand this equivalence.

That $forall x in {x_0, x_1 ldots x_n} P(x)$ means $P(x_0) land P(x_1) ldots P(x_n)$ looks pretty straighfoward to me. However, how the formula $forall x P(x)$, without expliciting the quantification domain, can be expressed in propositional logic as a chain of logical conjunctions still remains a mistery to me. $forall x P(x)$ is equivalent to $forall x in {x} P(x)$ and to $forall x in {x, x_0, x_1 ldots x_n} P(x)$. The fact that at least one element in the domain of discourse must be equal to the bound variable $x$ makes impossible to eliminate the universal quantifier from the formula $forall xP(x)$ and, thus, to convert it to propositional logic.

There would be the possibility of defining the universal set $U = {x : x=x}$ and then stating that $forall x P(x)$ is the same as $forall x in U P(x)$. In this case, since the universal set contains all objects, it would be impossible to limit the universe of discourse to a defined set of elements $x_0, x_1 ldots x_n$ and state that formula $forall x P(x)$ is equivalent to $P(x_0) land P(x_1) ldots P(x_n)$; it could be equivalent, for example, to $P(y_0) land P(y_1) ldots P(y_n)$ or to a chain of predicates about any other variables.