Symbol to write 'for all $x$ that satisfy: $|x-2|\lt c$'

Question

For example in the epsilon delta definition that states: a limit $L$ exists when $x$ approaches $a\iff (\forall \epsilon\gt 0) (\exists\delta\gt 0)$ for all $x$ that satisfy $0\lt|x-a|\lt\delta\implies|f(x)-L|\lt\epsilon$

How can I write this without saying "for all $x$ that satisfy"?

Answer 1

This is classically written as

$$(\forall \epsilon > 0)(\exists \delta > 0)(\forall x)[0 < |x - a| < \delta \Rightarrow |f(x) - L| < \epsilon]$$