Theorem injust

Description: Soundness justification theorem for df-in . (Contributed by Rodolfo Medina, 28-Apr-2010) (Proof shortened by Andrew Salmon, 9-Jul-2011)

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