Can I use ''exist'' predicate in the definition of algebra in equationally definable class?

Question

I'm reading Algebraic Theories by Ernest G. Manes,

and I'm wondering about equationally-definable class.

For example, group is a typical example of equationally-definable algebra,

but the definition of group uses ''for-all'' sentence: $\forall x,y,z\in X.\;x\cdot(y\cdot z)=(x\cdot y)\cdot z$;

thus I'm curious that whether can I use $\forall$, $\exists$, etc. in definition of equationally-definable algebra.