Theorem elab6g

Description: Membership in a class abstraction. Class version of sb6 . (Contributed by SN, 5-Oct-2024)

	Ref	Expression
Assertion	elab6g	|- ( A e. V -> ( A e. { x | ph } <-> A. x ( x = A -> ph ) ) )

Proof

Step	Hyp	Ref	Expression
1		eleq1	|- ( y = A -> ( y e. { x | ph } <-> A e. { x | ph } ) )
2		eqeq2	|- ( y = A -> ( x = y <-> x = A ) )
3	2	imbi1d	|- ( y = A -> ( ( x = y -> ph ) <-> ( x = A -> ph ) ) )
4	3	albidv	|- ( y = A -> ( A. x ( x = y -> ph ) <-> A. x ( x = A -> ph ) ) )
5		df-clab	|- ( y e. { x | ph } <-> [ y / x ] ph )
6		sb6	|- ( [ y / x ] ph <-> A. x ( x = y -> ph ) )
7	5 6	bitri	|- ( y e. { x | ph } <-> A. x ( x = y -> ph ) )
8	1 4 7	vtoclbg	|- ( A e. V -> ( A e. { x | ph } <-> A. x ( x = A -> ph ) ) )