# Existential instantiation

Type Rule of inference Predicate logic

In predicate logic, existential instantiation (also called existential elimination)[1][2][3] is a rule of inference which says that, given a formula of the form $\displaystyle{ (\exists x) \phi(x) }$, one may infer $\displaystyle{ \phi(c) }$ for a new constant symbol c. The rule has the restrictions that the constant c introduced by the rule must be a new term that has not occurred earlier in the proof, and it also must not occur in the conclusion of the proof. It is also necessary that every instance of $\displaystyle{ x }$ which is bound to $\displaystyle{ \exists x }$ must be uniformly replaced by c. This is implied by the notation $\displaystyle{ P\left({a}\right) }$, but its explicit statement is often left out of explanations.

In one formal notation, the rule may be denoted by

$\displaystyle{ \exists x P \left({x}\right) \implies P \left({a}\right) }$

where a is a new constant symbol that has not appeared in the proof.