The above rules state that if the formula above the line is a theorem formally deducted from axioms by application of the syllogisms, then
the formula below the line is also a formal theorem. Usually, some finite set of identities is given as axiom
schemata.

Equational logic can be combined with first-order logic. In this case, the fourth rule is extended onto predicate symbols as well,
and the fifth rule is omitted. These syllogisms can
be turned into axiom schemata having the form of
implications to which Modus
Ponens can be applied. Major results of first-order
logic hold in this extended theory.

If every identity in
is viewed as two rewrite rules transforming the left-hand side into the right-hand
side and vice versa, then the respective term
rewriting system is equivalent to the equational logic defined by : The identity is deducible in the equational
logic iff in the term
rewriting system. This property is called logicality of term
rewriting systems.

Equational logic is complete, since if algebra is a model for , i.e., all identities from hold in algebra (cf. universal algebra),
then
holds in iff it can be deduced in the equational logic defined by . This theorem is sometimes known as
Birkhoff's theorem.