Theorem injust 3179

Description: Soundness justification theorem for df-in 3180. (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 2270	. . . 4 |- ( x = z -> ( x e. A <-> z e. A ) )
2		eleq1 2270	. . . 4 |- ( x = z -> ( x e. B <-> z e. B ) )
3	1, 2	anbi12d 473	. . 3 |- ( x = z -> ( ( x e. A /\ x e. B ) <-> ( z e. A /\ z e. B ) ) )
4	3	cbvabv 2332	. 2 |- { x | ( x e. A /\ x e. B ) } = { z | ( z e. A /\ z e. B ) }
5		eleq1 2270	. . . 4 |- ( z = y -> ( z e. A <-> y e. A ) )
6		eleq1 2270	. . . 4 |- ( z = y -> ( z e. B <-> y e. B ) )
7	5, 6	anbi12d 473	. . 3 |- ( z = y -> ( ( z e. A /\ z e. B ) <-> ( y e. A /\ y e. B ) ) )
8	7	cbvabv 2332	. 2 |- { z | ( z e. A /\ z e. B ) } = { y | ( y e. A /\ y e. B ) }
9	4, 8	eqtri 2228	1 |- { x | ( x e. A /\ x e. B ) } = { y | ( y e. A /\ y e. B ) }

Syntax hints:   /\ wa 104   = wceq 1373   e. wcel 2178   {cab 2193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203

This theorem is referenced by: (None)