Universal instantiation

In predicate logic, universal instantiation[1][2][3] (UI; also called universal specification or universal elimination, and sometimes confused with dictum de omni) is a valid rule of inference from a truth about each member of a class of individuals to the truth about a particular individual of that class. It is generally given as a quantification rule for the universal quantifier but it can also be encoded in an axiom schema. It is one of the basic principles used in quantification theory.

Example: "All dogs are mammals. Fido is a dog. Therefore Fido is a mammal."

In symbols, the rule as an axiom schema is

${\displaystyle \forall x\,A\Rightarrow A\{x\mapsto a\},}$

for every formula A and every term a, where ${\displaystyle A\{x\mapsto a\}}$ is the result of substituting a for each free occurrence of x in A. ${\displaystyle \,A\{x\mapsto a\}}$ is an instance of ${\displaystyle \forall x\,A.}$

And as a rule of inference it is

from ⊢ ∀x A infer ⊢ A{x↦a}.

Irving Copi noted that universal instantiation "...follows from variants of rules for 'natural deduction', which were devised independently by Gerhard Gentzen and Stanisław Jaśkowski in 1934." [4]