Theorem injust 3040

Description: Soundness justification theorem for df-in 3041. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Assertion

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

Distinct variable groups:   x, A   x, B   y, A   y, B

Proof of Theorem injust

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2175	. . . 4 |- ( x = z -> ( x e. A <-> z e. A ) )
2		eleq1 2175	. . . 4 |- ( x = z -> ( x e. B <-> z e. B ) )
3	1, 2	anbi12d 462	. . 3 |- ( x = z -> ( ( x e. A /\ x e. B ) <-> ( z e. A /\ z e. B ) ) )
4	3	cbvabv 2236	. 2 |- { x | ( x e. A /\ x e. B ) } = { z | ( z e. A /\ z e. B ) }
5		eleq1 2175	. . . 4 |- ( z = y -> ( z e. A <-> y e. A ) )
6		eleq1 2175	. . . 4 |- ( z = y -> ( z e. B <-> y e. B ) )
7	5, 6	anbi12d 462	. . 3 |- ( z = y -> ( ( z e. A /\ z e. B ) <-> ( y e. A /\ y e. B ) ) )
8	7	cbvabv 2236	. 2 |- { z | ( z e. A /\ z e. B ) } = { y | ( y e. A /\ y e. B ) }
9	4, 8	eqtri 2133	1 |- { x | ( x e. A /\ x e. B ) } = { y | ( y e. A /\ y e. B ) }

Syntax hints:   /\ wa 103   = wceq 1312   e. wcel 1461   {cab 2099

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109

This theorem is referenced by: (None)