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.