From wikipedia [quantification][1] has meaning:

> In logic, quantification is the
> binding of a variable ranging over a
> domain of discourse

**Is there any formal "definition" of universal quantifier for example using definition of domain of discourse?** 

I mean a formula build without universal quantifier, and existential one which has the same meaning if referenced to defined domain of discourse?

For example:
Suppose we use domain of discourse (DoD) given by sentence $ U = \{ x|\phi(x) \}$ for some $\phi(x)$. Then naively we may wrote:

($\forall (x \in U)   \Phi(x) ) \equiv  ( \{ x|\phi(x) \} => \Phi(x) )$

In words: to say that some property follows for every x in DoD is the same as to say that if x is chosen from DoD then has this property.


We may try also the folowing one:
($\forall (x \in U)   \Phi(x) ) \equiv (( \{ x|\phi(x) \} => \Phi(x) ) => (\phi(x) <=> \Phi(x) ))$

In words: to say that some property follows for every x in DoD is the same as to say that $\phi$ and $\Phi$ are evenly spanned. 

Do  You know any reference for such matter?

  [1]: http://en.wikipedia.org/wiki/Quantification