Theorem bj-clel3gALT

Description: Alternate proof of clel3g . (Contributed by BJ, 1-Sep-2024) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	bj-clel3gALT	|- ( B e. V -> ( A e. B <-> E. x ( x = B /\ A e. x ) ) )

Proof

Step	Hyp	Ref	Expression
1		elisset	|- ( B e. V -> E. x x = B )
2	1	biantrurd	|- ( B e. V -> ( A e. B <-> ( E. x x = B /\ A e. B ) ) )
3		19.41v	|- ( E. x ( x = B /\ A e. B ) <-> ( E. x x = B /\ A e. B ) )
4	2 3	bitr4di	|- ( B e. V -> ( A e. B <-> E. x ( x = B /\ A e. B ) ) )
5		eleq2	|- ( x = B -> ( A e. x <-> A e. B ) )
6	5	bicomd	|- ( x = B -> ( A e. B <-> A e. x ) )
7	6	pm5.32i	|- ( ( x = B /\ A e. B ) <-> ( x = B /\ A e. x ) )
8	7	exbii	|- ( E. x ( x = B /\ A e. B ) <-> E. x ( x = B /\ A e. x ) )
9	4 8	bitrdi	|- ( B e. V -> ( A e. B <-> E. x ( x = B /\ A e. x ) ) )