Definition 12: Given statements \phi and \psi and a variable x , we may refer to the universally quantified implication \forall x (\phi \Rightarrow \psi), \hspace{5mm} \textup{i.e.} \hspace{5mm} \textup{"for all } x \textup{, we have that } \phi \textup{ implies } \psi \textup{"}, as a bounded universal statement. We may also express this by saying \textup{"for all } x \textup{ such that } \phi \textup{, we have } \psi \textup{"}. The generalized universal statement \forall a, b, ..., z (\phi \wedge \psi) is called a bounded generalized universal statement, or simply a bounded universal statement if the context is clear, and may be expressed as \textup{"for all } a, b, ..., z \textup{ such that } \phi \textup{, we have } \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 \forall x (\phi \Rightarrow \psi) by \textup{"for all } \mathsf{D}\textup{'s } x \textup{, we have } \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 \forall a, b, ..., z \big ( (\phi_a \wedge \phi_b \wedge \cdots \wedge \phi_z \wedge) \Rightarrow \psi \big) as \textup{"for all } \mathsf{A}\textup{'s} \textup{ } a, \mathsf{B}\textup{'s} \textup{ } b, ..., \textup{ and } \mathsf{Z}\textup{'s} \textup{ } z \textup{, we have } \psi \textup{"}, where we made use of generalized conjunction.