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 existence statement. We may write “is” in place of “exists”, and may add “some” or “at least one” after these; we may also say “with” instead of “such that”. The notation \exists x \phi may also be used. 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 a generalized 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.