What is the symmetry between the definitions of the bounded universal/existential quantifiers? What is the symmetry between the definitions of the bounded universal/existential quantifiers?
$\forall x \in A, B(x)$ means $\forall x (x \in A \rightarrow B(x))$
$\exists x \in A, B(x)$ means $\exists x (x \in A \land B(x))$
These make intuitive sense, but I would expect there to be some kind of symmetry between how the definitions of the bounded quantifiers work, and I can't see one. $A \rightarrow B$ means $\lnot A \lor B$ which doesn't seem to have a direct relationship with $A \land B$. What am I missing? 

Is this more symmetrical?
$\forall x \in A, B(x)$ means $\forall x (x \notin A \lor B(x))$
$\exists x \in A, B(x)$ means $\exists x (x \in A \land B(x))$
