Database: A variable not occurring in a statement does not occur quantified in the statement

Corollary 31: If a variable $x$ does not occur in a statement $\phi$ , then $x$ does not occur quantified in $\phi$ .

Proof: We know by [30] that $x$ occurs in $\phi$ precisely if it occurs free or occurs quantified, or both, in $\phi$ . Thus, if $x$ does not occur in $\phi$ then it cannot possibly occur quantified in $\phi$ either. $\square$

This database entry builds on the following:

Terminology: Class, statement
Definition: Occurrence of a variable
Definition: Free occurrence of a variable
Definition: Quantified occurrence of a variable
Lemma: A variable occurs in a statement precisely if it occurs free, occurs quantified, or both