Existential instantiation

In predicate logic, existential instantiation (also called existential elimination)  is a rule of inference which says that, given a formula of the form $$(\exists x) \phi(x)$$, one may infer $$\phi(c)$$ 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 $$x$$ which is bound to $$\exists x$$ must be uniformly replaced by c. This is implied by the notation $$P\left({a}\right)$$, but its explicit statement is often left out of explanations.

In one formal notation, the rule may be denoted by
 * $$\exists x P \left({x}\right) \implies P \left({a}\right)$$

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