Existential Generalisation

Context
Predicate Logic.

This is an extension of the Rule of Addition as follows:

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

If 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 $$P$$.

Some authors call this the Rule of Existential Introduction.

Proof
The propositional expansion of $$\exists x: P \left({x}\right)$$ is:

$$P \left({\mathbf{X}_1}\right) \lor P \left({\mathbf{X}_2}\right) \lor P \left({\mathbf{X}_3}\right) \lor \ldots$$

We have the fact that $$P \left({\mathbf{a}}\right)$$ where $$\mathbf{a}$$ is one of the above $$\mathbf{X}_1, \mathbf{X}_2, \mathbf{X}_3, \ldots$$, as it is by definition.

So the above statement $$P \left({\mathbf{X}_1}\right) \lor P \left({\mathbf{X}_2}\right) \lor P \left({\mathbf{X}_3}\right) \lor \ldots$$ follows by extension of the rule of addition.