To show constructively that a theory has quantifier elimination, it suffices to show that we can eliminate an existential quantifier applied to a conjunction of literals, that is, show that each formula of the form: \[

 \exists x. \bigwedge_{i=1}^n L_i

\] where each is a literal, is equivalent to a quantifier-free formula. Indeed, suppose we know how to eliminate quantifiers from conjunctions of formulae, then if is a quantifier-free formula, we can write it in disjunctive normal form \[

 \bigvee_{j=1}^m \bigwedge_{i=1}^n L_{ij}

\] and use the fact that \[

 \exists x. \bigvee_{j=1}^m \bigwedge_{i=1}^n L_{ij}

\] is equivalent to \[

 \bigvee_{j=1}^m \exists x. \bigwedge_{i=1}^n L_{ij}

\]

Finally, to eliminate a universal quantifier \[

 \forall x. F

\] where is quantifier-free, we transform into disjunctive normal form, and use the fact that is equivalent to

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