Theorem difjust

Description: Soundness justification theorem for df-dif . (Contributed by Rodolfo Medina, 27-Apr-2010) (Proof shortened by Andrew Salmon, 9-Jul-2011)

	Ref	Expression
Assertion	difjust	|- { x | ( x e. A /\ -. x e. B ) } = { y | ( y e. A /\ -. y e. B ) }

Proof

Step	Hyp	Ref	Expression
1		eleq1w	|- ( x = z -> ( x e. A <-> z e. A ) )
2		eleq1w	|- ( x = z -> ( x e. B <-> z e. B ) )
3	2	notbid	|- ( x = z -> ( -. x e. B <-> -. z e. B ) )
4	1 3	anbi12d	|- ( x = z -> ( ( x e. A /\ -. x e. B ) <-> ( z e. A /\ -. z e. B ) ) )
5	4	cbvabv	|- { x | ( x e. A /\ -. x e. B ) } = { z | ( z e. A /\ -. z e. B ) }
6		eleq1w	|- ( z = y -> ( z e. A <-> y e. A ) )
7		eleq1w	|- ( z = y -> ( z e. B <-> y e. B ) )
8	7	notbid	|- ( z = y -> ( -. z e. B <-> -. y e. B ) )
9	6 8	anbi12d	|- ( z = y -> ( ( z e. A /\ -. z e. B ) <-> ( y e. A /\ -. y e. B ) ) )
10	9	cbvabv	|- { z | ( z e. A /\ -. z e. B ) } = { y | ( y e. A /\ -. y e. B ) }
11	5 10	eqtri	|- { x | ( x e. A /\ -. x e. B ) } = { y | ( y e. A /\ -. y e. B ) }