Von Neumann's theorem

Statement
Suppose we have a variety of algebras: a collection of algebras with a fixed signature (i.e., a collection of operations of fixed arities), that is closed under taking subalgebras, quotients and arbitrary direct products.

Then the given variety is an equational variety: there exists a collection of equations such that an algebra with that signature is in the variety if and only if each of those equations is satisfied for all elements in the algebra.