Theorem sbalex

Description: 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 ph . (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)

	Ref	Expression
Assertion	sbalex	$$⊢\exists x\left(x=t\wedge \phi \right)\leftrightarrow \forall x\left(x=t\to \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		nfe1	$$⊢Ⅎx\exists x\left(x=t\wedge \phi \right)$$
2		ax12ev2	$$⊢\exists x\left(x=t\wedge \phi \right)\to \left(x=t\to \phi \right)$$
3	1 2	alrimi	$$⊢\exists x\left(x=t\wedge \phi \right)\to \forall x\left(x=t\to \phi \right)$$
4		equs4v	$$⊢\forall x\left(x=t\to \phi \right)\to \exists x\left(x=t\wedge \phi \right)$$
5	3 4	impbii	$$⊢\exists x\left(x=t\wedge \phi \right)\leftrightarrow \forall x\left(x=t\to \phi \right)$$