How does the statement that somebody envies everybody imply that everybody is envied by somebody?

In formal logic, the statement that "somebody envies everybody" and the statement that "everybody is envied by somebody" are not logically equivalent; they represent different relationships based on the scope of the quantifiers involved.

Using the predicate calculus, let $\varphi xy$ represent the relation "$x$ envies $y$."

• "Somebody envies everybody" is represented as $(\exists x)(\forall y)\varphi xy$. This asserts the existence of at least one specific individual who envies every person in the domain.1
• "Everybody is envied by somebody" is represented as $(\forall y)(\exists x)\varphi xy$. This asserts that for every person $y$, there exists at least one person $x$ who envies them.1

The first statement implies the second, because if there is one person who envies everyone, then every person is indeed envied by that specific individual. However, the reverse is not true: it is possible for everyone to be envied by someone without there being a single "universal envier" who envies the entire group. Therefore, the implication only holds in one direction.12