Theorem sbalexOLD

Description: Obsolete version of sbalex as of 14-Aug-2025. (Contributed by NM, 14-Apr-2008) (Revised by BJ, 20-Dec-2020) (Revised by BJ, 21-Sep-2024) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		nfa1	$$⊢Ⅎx\forall x\left(x=t\to \phi \right)$$
2		ax12v2	$$⊢x=t\to \left(\phi \to \forall x\left(x=t\to \phi \right)\right)$$
3	2	imp	$$⊢\left(x=t\wedge \phi \right)\to \forall x\left(x=t\to \phi \right)$$
4	1 3	exlimi	$$⊢\exists x\left(x=t\wedge \phi \right)\to \forall x\left(x=t\to \phi \right)$$
5		equs4v	$$⊢\forall x\left(x=t\to \phi \right)\to \exists x\left(x=t\wedge \phi \right)$$
6	4 5	impbii	$$⊢\exists x\left(x=t\wedge \phi \right)\leftrightarrow \forall x\left(x=t\to \phi \right)$$