State inference rule for universal instantiation.

The universal instantiation inference rule in first-order logic allows us to derive a specific statement from a universal statement. It is formally stated as:

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Rule: If $\forall xP(x)$ is true, then for any specific term $a$, $P(a)$ is also true.

Symbolic Form: $$\frac{\forall xP(x)}{\therefore P(c)}$$ where $c$ is a constant, variable, or term.

Example: If $\forall x(x+2>5)$ is true, then we can instantiate it for $x=4$ to conclude $4+2>5$.

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This rule is valid because it preserves truth: if all elements satisfy $P(x)$, then any particular element must satisfy it.