Universal Instantiation/Informal Statement

Theorem
Suppose we have a universal statement:
 * $\forall x: \map P x$

where $\forall$ is the universal quantifier and $\map P x$ is a propositional function.

Then we can deduce:
 * $\map P {\mathbf a}$

where $\mathbf a$ is any arbitrary object we care to choose in the universe of discourse.

In natural language:


 * Suppose $P$ is true of everything in the universe of discourse.


 * ''Let $\mathbf a$ be an element of the universe of discourse."


 * Then $P$ is true of $\mathbf a$.

Proof
In the language of symbolic logic: