Existential Generalisation/Informal Statement

Theorem

 * $P \left({\mathbf a}\right) \vdash \exists x: P \left({x}\right)$

Suppose we have the following:
 * We can find an arbitrary object $\mathbf a$ in our universe of discourse which has the property $P$.

Then we may infer that:
 * there exists in that universe at least one object $x$ which has that property $P$.

This is called the Rule of Existential Generalisation and often appears in a proof with its abbreviation EG.