Definition 10: Given statements \phi and \psi and a variable x , we may refer to the existentially quantified conjunction \exists x (\phi \wedge \psi), \hspace{5mm} \textup{i.e.} \hspace{5mm} \textup{"there exists } x \textup{ such that } \phi \textup{ and } \psi \textup{"}, as a bounded existence statement. We might also express this as \textup{"there exists } x \textup{ with } \phi \textup{, such that } \psi \textup{"}. The generalized existence statement \exists a, b, ..., z (\phi \wedge \psi) is called a bounded generalized existence statement, or simply a bounded existence statement if the context is clear, and may be expressed as \textup{"there exists } a, b, ..., z \textup{ with } \phi \textup{ such that } \psi \textup{"},or similarly. Given some definition which specifies a shorthand of the form “ x is a \mathsf{D} ” for a statement \phi , we may express \exists x (\phi \wedge \psi) by saying \textup{"there exists a } \mathsf{D} \textup{ } x \textup{ such that } \psi \textup{"}. Similarly, given definitions \mathsf{A}, \mathsf{Z}, ..., \mathsf{Z} specifying shorthands of the form “ a is an \mathsf{A} “, b is a \mathsf{B} , …, z is a \mathsf{Z} ” for statements \phi_a, \phi_b, ..., \phi_z respectively, we may express \exists a, b, ..., z (\phi_a \wedge \phi_b \wedge \cdots \wedge \phi_z \wedge \psi) as \textup{"there exists an } \mathsf{A} \textup{ } a, \textup{ a } \mathsf{B} \textup{ } b, ..., \textup{ and a } \mathsf{Z} \textup{ } z \textup{ such that } \psi \textup{"}, where we made use of generalized conjunction.