# 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.

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:

 $\ds \forall x: \,$ $\ds \map P x$  $\ds$ $\ds \therefore \ \$ $\ds \map P {\mathbf a}$  $\ds$

$\blacksquare$