Theorem difjust 3158

Description: Soundness justification theorem for df-dif 3159. (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 2259	. . . 4 |- ( x = z -> ( x e. A <-> z e. A ) )
2		eleq1 2259	. . . . 5 |- ( x = z -> ( x e. B <-> z e. B ) )
3	2	notbid 668	. . . 4 |- ( x = z -> ( -. x e. B <-> -. z e. B ) )
4	1, 3	anbi12d 473	. . 3 |- ( x = z -> ( ( x e. A /\ -. x e. B ) <-> ( z e. A /\ -. z e. B ) ) )
5	4	cbvabv 2321	. 2 |- { x | ( x e. A /\ -. x e. B ) } = { z | ( z e. A /\ -. z e. B ) }
6		eleq1 2259	. . . 4 |- ( z = y -> ( z e. A <-> y e. A ) )
7		eleq1 2259	. . . . 5 |- ( z = y -> ( z e. B <-> y e. B ) )
8	7	notbid 668	. . . 4 |- ( z = y -> ( -. z e. B <-> -. y e. B ) )
9	6, 8	anbi12d 473	. . 3 |- ( z = y -> ( ( z e. A /\ -. z e. B ) <-> ( y e. A /\ -. y e. B ) ) )
10	9	cbvabv 2321	. 2 |- { z | ( z e. A /\ -. z e. B ) } = { y | ( y e. A /\ -. y e. B ) }
11	5, 10	eqtri 2217	1 |- { x | ( x e. A /\ -. x e. B ) } = { y | ( y e. A /\ -. y e. B ) }

Syntax hints:   -. wn 3   /\ wa 104   = wceq 1364   e. wcel 2167   {cab 2182

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-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192

This theorem is referenced by: (None)