Fix an algebraic signature $\Omega$. Let $F$ be a finite set of equations in the signature $\Omega$, and let $I$ be an infinite set of equations in the signature $\Omega$ which generate precisely the same equational theory as the theory generated by $F$. Must there be a finite subset $F'$ of $I$ which generates the same equational theory as the theory generated by $F$?
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Yes. Since $F$ and $I$ are each equational theories with the same set of equational consequences, they each include each other's elements as consequences. Since equational logic (and much more of course) is compact, there is some finite subset of $I$ which suffices to prove all the finitely many equations in $F$.
(Note that this is an instance of a much more general phenomenon: whenever we have a compact logic such as equational logic or even full $\mathsf{FOL}$, every infinite axiomatization of a finitely axiomatizable theory in that logic can be reduced to a finite sub-axiomatization. This is somewhat related to my answer to your older question.)
Noah Schweber
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