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A representation by a set of first-order clauses, interpreted as their logical conjunction. General first-order formulas can be transformed by Skolemization and conversion to conjunctive normal form, preserving satisfiability in an expanded first-order signature.

In an expanded signature, Skolemization preserves equisatisfiability rather than equivalence under arbitrary interpretations of the new function symbols.

The usual transformation to clausal normal form preserves $\boxed{\text{equisatisfiability, but not necessarily logical equivalence}}$. Removing implications and distributing logical conjunction and logical disjunction preserve logical equivalence. Skolemization introduces new symbols and chooses witnesses as functions of preceding universally quantified variables.

For example,On the domain $\{0,1\}$, let $R$ be equality. Then $\phi$ is true. Expand this same first-order structure by interpreting $f$ as the constant function with value $0$. The resulting first-order structure falsifies $\psi$ at $x=1$. Interpreting $f$ as the identity instead satisfies $\psi$.

Thus every model of $\psi$ has a reduct satisfying $\phi$, and a model of $\phi$ can be expanded with suitable witnesses to satisfy $\psi$. That model-expansion property gives equisatisfiability, not logical equivalence for every interpretation of the added function. If “CNF” means a conjunctive matrix with existential quantifiers retained, Skolemization has not yet happened and an equivalent form is possible; this is why the convention must be specified.