Theorem elOLD

Description: Obsolete version of el as of 6-Apr-2026. (Contributed by NM, 4-Jan-2002) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	elOLD	|- E. y x e. y

Proof

Step	Hyp	Ref	Expression
1		ax-pr	|- E. y A. z ( ( z = x \/ z = x ) -> z e. y )
2		pm4.25	|- ( z = x <-> ( z = x \/ z = x ) )
3	2	imbi1i	|- ( ( z = x -> z e. y ) <-> ( ( z = x \/ z = x ) -> z e. y ) )
4	3	albii	|- ( A. z ( z = x -> z e. y ) <-> A. z ( ( z = x \/ z = x ) -> z e. y ) )
5		elequ1	|- ( z = x -> ( z e. y <-> x e. y ) )
6	5	equsalvw	|- ( A. z ( z = x -> z e. y ) <-> x e. y )
7	4 6	bitr3i	|- ( A. z ( ( z = x \/ z = x ) -> z e. y ) <-> x e. y )
8	7	exbii	|- ( E. y A. z ( ( z = x \/ z = x ) -> z e. y ) <-> E. y x e. y )
9	1 8	mpbi	|- E. y x e. y