Theorem injust 2978

Description: Soundness justification theorem for df-in 2979. (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 2141	. . . 4 |- ( x = z -> ( x e. A <-> z e. A ) )
2		eleq1 2141	. . . 4 |- ( x = z -> ( x e. B <-> z e. B ) )
3	1, 2	anbi12d 456	. . 3 |- ( x = z -> ( ( x e. A /\ x e. B ) <-> ( z e. A /\ z e. B ) ) )
4	3	cbvabv 2202	. 2 |- { x | ( x e. A /\ x e. B ) } = { z | ( z e. A /\ z e. B ) }
5		eleq1 2141	. . . 4 |- ( z = y -> ( z e. A <-> y e. A ) )
6		eleq1 2141	. . . 4 |- ( z = y -> ( z e. B <-> y e. B ) )
7	5, 6	anbi12d 456	. . 3 |- ( z = y -> ( ( z e. A /\ z e. B ) <-> ( y e. A /\ y e. B ) ) )
8	7	cbvabv 2202	. 2 |- { z | ( z e. A /\ z e. B ) } = { y | ( y e. A /\ y e. B ) }
9	4, 8	eqtri 2101	1 |- { x | ( x e. A /\ x e. B ) } = { y | ( y e. A /\ y e. B ) }

Syntax hints:   /\ wa 102   = wceq 1284   e. wcel 1433   {cab 2067

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077

This theorem is referenced by: (None)