Theorem elabd2

Description: Membership in a class abstraction, using implicit substitution. Deduction version of elab . (Contributed by GG, 12-Oct-2024) (Revised by BJ, 16-Oct-2024)

	Ref	Expression
Hypotheses	elabd2.ex	|- ( ph -> A e. V )
	elabd2.eq	|- ( ph -> B = { x | ps } )
	elabd2.is	|- ( ( ph /\ x = A ) -> ( ps <-> ch ) )
Assertion	elabd2	|- ( ph -> ( A e. B <-> ch ) )

Proof

Step	Hyp	Ref	Expression
1		elabd2.ex	|- ( ph -> A e. V )
2		elabd2.eq	|- ( ph -> B = { x | ps } )
3		elabd2.is	|- ( ( ph /\ x = A ) -> ( ps <-> ch ) )
4	2	eleq2d	|- ( ph -> ( A e. B <-> A e. { x | ps } ) )
5		elab6g	|- ( A e. V -> ( A e. { x | ps } <-> A. x ( x = A -> ps ) ) )
6	4 5	sylan9bb	|- ( ( ph /\ A e. V ) -> ( A e. B <-> A. x ( x = A -> ps ) ) )
7		elisset	|- ( A e. V -> E. x x = A )
8	3	pm5.74da	|- ( ph -> ( ( x = A -> ps ) <-> ( x = A -> ch ) ) )
9	8	albidv	|- ( ph -> ( A. x ( x = A -> ps ) <-> A. x ( x = A -> ch ) ) )
10		19.23v	|- ( A. x ( x = A -> ch ) <-> ( E. x x = A -> ch ) )
11	9 10	bitrdi	|- ( ph -> ( A. x ( x = A -> ps ) <-> ( E. x x = A -> ch ) ) )
12		pm5.5	|- ( E. x x = A -> ( ( E. x x = A -> ch ) <-> ch ) )
13	11 12	sylan9bb	|- ( ( ph /\ E. x x = A ) -> ( A. x ( x = A -> ps ) <-> ch ) )
14	7 13	sylan2	|- ( ( ph /\ A e. V ) -> ( A. x ( x = A -> ps ) <-> ch ) )
15	6 14	bitrd	|- ( ( ph /\ A e. V ) -> ( A e. B <-> ch ) )
16	1 15	mpdan	|- ( ph -> ( A e. B <-> ch ) )