Existential instantiation

Existential instantiation

Type	Rule of inference
Field	Predicate logic
Symbolic statement	${\displaystyle \exists xP\left({x}\right)\implies P\left({a}\right)}$

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

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

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