Universal Instantiation

Context
Predicate Logic.

Theorem
Suppose we have a universal statement: $$\forall x: P \left({x}\right)$$, where $$\forall$$ is the universal quantifier and $$P \left({x}\right)$$ is a propositional function.

Then we can deduce $$P \left({\mathbf{a}}\right)$$ where $$\mathbf{a}$$ is any arbitrary object we care to choose in the universe of discourse.

Proof
We can express $$\forall x$$ using its propositional expansion:

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

where $$\mathbf{X}_1, \mathbf{X}_2, \mathbf{X}_3 \ldots{}$$ is the complete set of the objects in the universe of discourse.

We can now apply the rule of simplification and the result follows.