# Universal Instantiation

## Theorem

### Informal Statement

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 $\mathbf a$.

### Universal Instantiation in Models

Let $\map {\mathbf A} x$ be a WFF of predicate logic.

Let $\tau$ be a term which is freely substitutable for $x$ in $\mathbf A$.

Then $\forall x: \map {\mathbf A} x \implies \map {\mathbf A} \tau$ is a tautology.

### Universal Instantiation in Proof Systems

Let $\map {\mathbf A} x$ be a WFF of predicate logic.

Let $\tau$ be a term which is freely substitutable for $x$ in $\mathbf A$.

Let $\mathscr H$ be Hilbert proof system instance 1 for predicate logic.

Then:

$\forall x: \map {\mathbf A} x \vdash_{\mathscr H} \map {\mathbf A} \tau$

is a provable consequence in $\mathscr H$.

## Also known as

Some authors call this the Rule of Universal Elimination and it is then abbreviated UE.