What is the formal definition of repeated limit using quantifiers?

The basic question is what has been asked in the title. I looked for the definition here, here and here but no definition uses quantifiers.

I tried to formulate the definition but succeeded only partially. For this I considered the following special case,

Let $f:D(\subseteq\mathbb{R}^2)\to\mathbb{R}$ and $(a,b)\in D$. Then we say $\displaystyle\lim_{x\to a}f(x,y)=L(y)$ if $$(\forall y)(\exists L_y\in\mathbb{R})\left[(\forall \varepsilon>0)(\exists \delta>0)\right[|x-a|<\delta\rightarrow |f(x,y)-L_y|<\varepsilon]]$$

• $\lim_{x\to a}f(x,y)$ makes sense, from the formal definition of a limit, as a function of $y$. In which case one just uses that function and the formal definition of a limit to define $\lim_{y\to b}(\lim_{x\to a}f(x,y))$. Is there anything about this that you don't understand? There is really no point in writing things with epsilons and deltas, since we have that for the definition of a single limit, and then we can just define iterated limits in words invoking the concept of a limit repeatedly. – anon Jun 29 '16 at 3:13
• @anon: I don't think that there is anything in the definition (at least in the links and in your phrasing of the definition) that I don't understand. I just want to know what a formal definition of the iterated limits will look like. – user 170039 Jun 29 '16 at 5:50