Existence and uniqueness results are fundamental in mathematics because we often define objects by their properties, and then start calling them "the object", which is fantastically convenient.
But calling something "the object" only makes sense if there exists exactly one, and only one, object that satisfies the properties.
One particular context where these come up very explicitly is in solutions to differential equations, e.g. existence and uniqueness of solutions of partial differential equations.
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Existential quantification forms . In a first-order structure, it is true at an assignment to when some domain element makes the instance true. In natural deduction, elimination uses a fresh variable for the hypothetical witness; that variable may not escape into the conclusion or the remaining undischarged assumptions.
Existential quantification is a concept from mathematical logic and predicate logic that expresses that there exists at least one element in a particular domain for which a certain property or predicate holds true. It is typically denoted using the symbol ∃ (the existential quantifier).