Quantifier elimination, $(\mathbb{R}, <)$

Question

I have a general question about quantifier elimination. Which kinds of formulas do you have to observe?

For example let T be the theory of $(\mathbb{R}, <)$ and I want to show, that this theory has quantifier elimination.

What do I have to do in general to proof it. Are there just specific formulas I have to observe? Does quantifier elimination always have the same steps, so you always observe the same formulas and show that they are equivalent to a quantifier free formula?

Thanks in advance.

Edit: Spelling improved.

Answer 1

In proving quantifier elimination, it suffices to treat formulas consisting of a single existential quantifier applied to a conjunction of atomic and negated atomic formulas. But that is, as far as I know, the only general reduction available. After that, you need to consider the particular theory and do some real work.