if sentences can be translated in a truth-preserving fashion into sentences of (functional) existential second-order logic (formula_368) by means of the skolemization procedure (see above). vice versa, every formula_368 can be translated into an if sentence by means of a variant of the walkoe-enderton translation procedure for partially-ordered quantifiers (). in other words, if logic and formula_368 are expressively equivalent at the level of sentences. this equivalence can be used to prove many of the properties that follow; they are inherited from formula_368 and in many cases similar to properties of fol.