The definition of uniform continuity is:
Given any $\varepsilon>0\ \exists\delta>0\ \forall x\in I \ \forall y\in I\ \left(\text{if }|x-y|<\delta\text{ then }\ |f(x)-f(y)|<\varepsilon\right)$ where $I=$ the interval on which $f$ is defined
so this means that $\delta$ must remain constant for a given $\epsilon$.
The negation statement is:
$\exists \varepsilon>0\ \forall\delta>0\ \exists x,y \in I\left(\text{if }|x-y|<\delta\text{ and}\ |f(x)-f(y)|\geq\varepsilon\right)$ where $I=$ the interval on which $f$ is defined.
I am trying to understand this logically without reverting to rules. The part which I don't understand is $\forall\delta>0$.