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A variable occurrence in a first-order formula is bound when it lies in the scope of a quantifier binding that name. The same name can have both bound and free occurrences in one formula. A first-order substitution replaces only free occurrences; a fresh renaming of bound occurrences and their binder preserves the formula up to alpha equivalence.

To perform a first-order substitution, first rename each conflicting bound variable to a fresh name, then replace the designated free occurrences. The fresh names must avoid the formula's relevant free names and all names in the inserted first-order term. For instance, substituting $f(u,t)$ for $z$ in $\exists tQ(z,t)$ requires renaming the binder, producing $\exists sQ(f(u,t),s)$ rather than $\exists tQ(f(u,t),t)$. The latter would change the intended meaning by capturing $t$.

The term $x+z$ is free for substitution for $z$. The only free occurrence of the target $z$ is in the outside disjunct $y<z$, beyond both quantifier scopes. The occurrences under $\forall z$ are bound variables and are not replaced. Therefore no renaming is required, andThe new $x$ and $z$ in the last disjunct are free. First-order substitution replaces the original free occurrences once; it does not recursively substitute into the inserted term.