Equivalent expressions
If X is a domain of x and P(x) is a predicate dependent on x, then the universal proposition is expressed in Boolean algebra terms as
$\forall x\in X, P(x) \equiv \{x\in X\} \rightarrow P(x) \equiv \{x\notin X\} \vee P(x)$,
which equivalently reads "if x is in X, then P(x) is true." If x is not in X, then P(x) is indeterminate. Note that the truth of the expression requires only that x be in X, so it can be any x in X, independent of P(x), whereas the falsity of the expression, or the truth of
$\{x\in X\} \wedge \neg P(x)$,
additionally requires that x be such that P(x) evaluates to false; this is the reason behind calling x a "bound variable." This last expression can thus be read as "for some x in X, P(x) is false," or "there exists an x in X such that P(x) is false." So, we now have the equivalent Boolean expression for the existential proposition:
$\exists x\in X : P(x) \equiv \{x\in X\} \wedge P(x)$.
See also:
Quantification