D is a domain of x and P(x) is a predicate dependent on object variable x, then the universal proposition can be expressed as

This notation is known as restricted or relativized or bounded quantification. Equivalently one can write,

The existential proposition can be expressed with bounded quantification as

or equivalently

Together with negation, only one of either the universal or existential quantifier is needed to perform both tasks:

which shows that to disprove