Equivalence of two ways to express proper substitution of a setvar for another setvar disjoint from it in a formula. This proof of their equivalence does not use [df-sb].

That both sides of the biconditional express proper substitution is proved by sb5 and sb6. The implication "to the left" is equs4v and does not require [ax-10] nor [ax-12]. It also holds without disjoint variable condition if we allow more axioms (see equs4 ). Theorem 6.2 of [Quine] p. 40. Theorem equs5 replaces the disjoint variable condition with a distinctor antecedent. Theorem equs45f replaces the disjoint variable condition on x , t with the nonfreeness hypothesis of \( t' \) in \( \varphi \). (Contributed by NM, 14-Apr-2008.) Revised to use equsexv in place of equsex in order to remove dependency on [ax-13]. (Revised by BJ, 20-Dec-2020.) Revise to remove dependency on [df-sb]. (Revised by BJ, 21-Sep-2024.) (Proof shortened by SN, 14-Aug-2025.)