Theorem clel2gOLD

Description: Obsolete version of clel2g as of 1-Sep-2024. (Contributed by NM, 18-Aug-1993) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	clel2gOLD	$$⊢A\in V\to \left(A\in B\leftrightarrow \forall x\left(x=A\to x\in B\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		nfv	$$⊢ℲxA\in B$$
2		eleq1	$$⊢x=A\to \left(x\in B\leftrightarrow A\in B\right)$$
3	1 2	ceqsalg	$$⊢A\in V\to \left(\forall x\left(x=A\to x\in B\right)\leftrightarrow A\in B\right)$$
4	3	bicomd	$$⊢A\in V\to \left(A\in B\leftrightarrow \forall x\left(x=A\to x\in B\right)\right)$$