Universal instantiation

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

Formally, the rule as an axiom schema is given as

${\displaystyle \mathrm{\forall }xA\Rightarrow A\{x\mapsto t\},}$

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

And as a rule of inference it is

from ${\displaystyle ⊢\mathrm{\forall }xA}$ infer ${\displaystyle ⊢A\{x\mapsto t\}.}$

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."

Quine

According to Willard Van Orman Quine, universal instantiation and existential generalization are two aspects of a single principle, for instead of saying that "∀x x = x" implies "Socrates = Socrates", we could as well say that the denial "Socrates ≠ Socrates" implies "∃x x ≠ x". The principle embodied in these two operations is the link between quantifications and the singular statements that are related to them as instances. Yet it is a principle only by courtesy. It holds only in the case where a term names and, furthermore, occurs referentially.