It is simple to see that the class of algebras satisfying a given set of equations will always be closed under the HSP operations, so the burden of Birkhoff's theorem is the converse: classes of algebras that satisfy those conditions must be equational.
The difficult part of Birkhoff's theorem is proving the converse: classes of algebras that are closed under the HSP operations must be equational. It is simple to see that the class of algebras satisfying a given set of equations will always be closed under the HSP operations.
It is simple to see that the class of algebras satisfying a given set of equations will always be closed under the HSP operations. The difficult part of Birkhoff's theorem is proving the converse: classes of algebras that are closed under the HSP operations must be equational.
It is simple to see that the class of algebras satisfying a given set of equations will always be closed under the HSP operations. The difficult part of Birkhoff's theorem is proving the converse: classes of algebras that are closed under the HSP operations must be equational.