Theorem difjust 3099

Description: Soundness justification theorem for df-dif 3100. (Contributed by Rodolfo Medina, 27-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Assertion

Ref	Expression
difjust	|- { 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 difjust

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2217	. . . 4 |- ( x = z -> ( x e. A <-> z e. A ) )
2		eleq1 2217	. . . . 5 |- ( x = z -> ( x e. B <-> z e. B ) )
3	2	notbid 657	. . . 4 |- ( x = z -> ( -. x e. B <-> -. z e. B ) )
4	1, 3	anbi12d 465	. . 3 |- ( x = z -> ( ( x e. A /\ -. x e. B ) <-> ( z e. A /\ -. z e. B ) ) )
5	4	cbvabv 2279	. 2 |- { x | ( x e. A /\ -. x e. B ) } = { z | ( z e. A /\ -. z e. B ) }
6		eleq1 2217	. . . 4 |- ( z = y -> ( z e. A <-> y e. A ) )
7		eleq1 2217	. . . . 5 |- ( z = y -> ( z e. B <-> y e. B ) )
8	7	notbid 657	. . . 4 |- ( z = y -> ( -. z e. B <-> -. y e. B ) )
9	6, 8	anbi12d 465	. . 3 |- ( z = y -> ( ( z e. A /\ -. z e. B ) <-> ( y e. A /\ -. y e. B ) ) )
10	9	cbvabv 2279	. 2 |- { z | ( z e. A /\ -. z e. B ) } = { y | ( y e. A /\ -. y e. B ) }
11	5, 10	eqtri 2175	1 |- { x | ( x e. A /\ -. x e. B ) } = { y | ( y e. A /\ -. y e. B ) }

Syntax hints:   -. wn 3   /\ wa 103   = wceq 1332   e. wcel 2125   {cab 2140

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150

This theorem is referenced by: (None)