Theorem bj-denoteslem

Description: Duplicate of issettru and bj-issettruALTV .

Lemma for bj-denotesALTV . (Contributed by BJ, 24-Apr-2024) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-denoteslem	|- ( E. x x = A <-> A e. { y | T. } )

Proof

Step	Hyp	Ref	Expression
1		vextru	|- x e. { y | T. }
2	1	biantru	|- ( x = A <-> ( x = A /\ x e. { y | T. } ) )
3	2	exbii	|- ( E. x x = A <-> E. x ( x = A /\ x e. { y | T. } ) )
4		dfclel	|- ( A e. { y | T. } <-> E. x ( x = A /\ x e. { y | T. } ) )
5	3 4	bitr4i	|- ( E. x x = A <-> A e. { y | T. } )