Universal specification moves from every object to one object. Universal generalization moves in the reverse direction: if a statement is proved for an arbitrary element, then it is true for every element. This rule is essential in theorem proofs, because most universal theorems cannot be proved by checking all cases. In this lesson, universal generalization in theorem proofs is developed through quantified arguments and symbolic proof patterns.

The Rule of Universal Generalization states that if an open statement $p(x)$ is proved for an arbitrarily chosen element of the universe, then the statement is true for every element of the universe. If $c$ is chosen arbitrarily from the universe and $p(c)$ is proved, then $\forall x\,p(x)$ is true.