Theorem clelab

Description: Membership of a class variable in a class abstraction. (Contributed by NM, 23-Dec-1993) (Proof shortened by Wolf Lammen, 16-Nov-2019) Avoid ax-11 , see sbc5ALT for more details. (Revised by SN, 2-Sep-2024)

	Ref	Expression
Assertion	clelab	|- ( A e. { x | ph } <-> E. x ( x = A /\ ph ) )

Proof

Step	Hyp	Ref	Expression
1		elissetv	|- ( A e. { x | ph } -> E. y y = A )
2		exsimpl	|- ( E. x ( x = A /\ ph ) -> E. x x = A )
3		iseqsetv-cleq	|- ( E. x x = A <-> E. y y = A )
4	2 3	sylib	|- ( E. x ( x = A /\ ph ) -> E. y y = A )
5		eleq1	|- ( y = A -> ( y e. { x | ph } <-> A e. { x | ph } ) )
6		df-clab	|- ( y e. { x | ph } <-> [ y / x ] ph )
7		sb5	|- ( [ y / x ] ph <-> E. x ( x = y /\ ph ) )
8	6 7	bitri	|- ( y e. { x | ph } <-> E. x ( x = y /\ ph ) )
9		eqeq2	|- ( y = A -> ( x = y <-> x = A ) )
10	9	anbi1d	|- ( y = A -> ( ( x = y /\ ph ) <-> ( x = A /\ ph ) ) )
11	10	exbidv	|- ( y = A -> ( E. x ( x = y /\ ph ) <-> E. x ( x = A /\ ph ) ) )
12	8 11	bitrid	|- ( y = A -> ( y e. { x | ph } <-> E. x ( x = A /\ ph ) ) )
13	5 12	bitr3d	|- ( y = A -> ( A e. { x | ph } <-> E. x ( x = A /\ ph ) ) )
14	13	exlimiv	|- ( E. y y = A -> ( A e. { x | ph } <-> E. x ( x = A /\ ph ) ) )
15	1 4 14	pm5.21nii	|- ( A e. { x | ph } <-> E. x ( x = A /\ ph ) )