Definition 9: Given a statement \phi and a variable x , the statement \textup{"there exists } x \textup{ such that } \phi \textup{"}is called an existentially quantified statement, or just existence statement. We may express such a statement using the notation \exists x \phi. Furthermore, we may use the following recursively defined notation:
(a) Base cases: We understand \exists a, b \phi to abbreviate \exists a \exists b \phi .
(b) Recursion: If \exists b, c, ..., z \phi is understood, we may further write \exists a, b, c, …, z \phi \hspace{3mm} \textup{for} \hspace{3mm} \exists a \exists b, c, …, z \phi.
A statement of this form is called an iterated existence statement, or just an existence statement if the context is clear. We may express \exists a, b, ..., z \phi as “there exists a, b, ..., z such that \phi “, or similar.