Informally : a formula A in prenex form containing existential quantifiers only is provable ( valid ) in first-order logic if and only if a disjunction composed of substitution instances of the quantifier-free subformula of A is a tautology ( propositionally derivable ).
22.
By construction, all these formulas are arithmetical ( no class variables are ever bound ) and, in fact, by putting the formula in Skolem prenex form one can see that every arithmetical formula is equivalent to a ? 0 " n " or ? 0 " n " formula for all large enough " n ".
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Next we consider a generic formula ? ( which no longer uses function or constant symbols ) and apply the prenex form theorem to find a formula ? in " normal form " such that ?a " ? ( ? being in " normal form " means that all the quantifiers in ?, if there are any, are found at the very beginning of ? ).
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