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Metamath Proof Explorer

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 ) }