Theorem preq12bg 3565

Description: Closed form of preq12b 3562. (Contributed by Scott Fenton, 28-Mar-2014.)

Assertion

Ref	Expression
preq12bg	|- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) ) -> ( { A , B } = { C , D } <-> ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) ) )

Proof of Theorem preq12bg

Dummy variables x y z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		preq1 3469	. . . . . . 7 |- ( x = A -> { x , y } = { A , y } )
2	1	eqeq1d 2089	. . . . . 6 |- ( x = A -> ( { x , y } = { z , D } <-> { A , y } = { z , D } ) )
3		eqeq1 2087	. . . . . . . 8 |- ( x = A -> ( x = z <-> A = z ) )
4	3	anbi1d 452	. . . . . . 7 |- ( x = A -> ( ( x = z /\ y = D ) <-> ( A = z /\ y = D ) ) )
5		eqeq1 2087	. . . . . . . 8 |- ( x = A -> ( x = D <-> A = D ) )
6	5	anbi1d 452	. . . . . . 7 |- ( x = A -> ( ( x = D /\ y = z ) <-> ( A = D /\ y = z ) ) )
7	4, 6	orbi12d 739	. . . . . 6 |- ( x = A -> ( ( ( x = z /\ y = D ) \/ ( x = D /\ y = z ) ) <-> ( ( A = z /\ y = D ) \/ ( A = D /\ y = z ) ) ) )
8	2, 7	bibi12d 233	. . . . 5 |- ( x = A -> ( ( { x , y } = { z , D } <-> ( ( x = z /\ y = D ) \/ ( x = D /\ y = z ) ) ) <-> ( { A , y } = { z , D } <-> ( ( A = z /\ y = D ) \/ ( A = D /\ y = z ) ) ) ) )
9	8	imbi2d 228	. . . 4 |- ( x = A -> ( ( D e. Y -> ( { x , y } = { z , D } <-> ( ( x = z /\ y = D ) \/ ( x = D /\ y = z ) ) ) ) <-> ( D e. Y -> ( { A , y } = { z , D } <-> ( ( A = z /\ y = D ) \/ ( A = D /\ y = z ) ) ) ) ) )
10		preq2 3470	. . . . . . 7 |- ( y = B -> { A , y } = { A , B } )
11	10	eqeq1d 2089	. . . . . 6 |- ( y = B -> ( { A , y } = { z , D } <-> { A , B } = { z , D } ) )
12		eqeq1 2087	. . . . . . . 8 |- ( y = B -> ( y = D <-> B = D ) )
13	12	anbi2d 451	. . . . . . 7 |- ( y = B -> ( ( A = z /\ y = D ) <-> ( A = z /\ B = D ) ) )
14		eqeq1 2087	. . . . . . . 8 |- ( y = B -> ( y = z <-> B = z ) )
15	14	anbi2d 451	. . . . . . 7 |- ( y = B -> ( ( A = D /\ y = z ) <-> ( A = D /\ B = z ) ) )
16	13, 15	orbi12d 739	. . . . . 6 |- ( y = B -> ( ( ( A = z /\ y = D ) \/ ( A = D /\ y = z ) ) <-> ( ( A = z /\ B = D ) \/ ( A = D /\ B = z ) ) ) )
17	11, 16	bibi12d 233	. . . . 5 |- ( y = B -> ( ( { A , y } = { z , D } <-> ( ( A = z /\ y = D ) \/ ( A = D /\ y = z ) ) ) <-> ( { A , B } = { z , D } <-> ( ( A = z /\ B = D ) \/ ( A = D /\ B = z ) ) ) ) )
18	17	imbi2d 228	. . . 4 |- ( y = B -> ( ( D e. Y -> ( { A , y } = { z , D } <-> ( ( A = z /\ y = D ) \/ ( A = D /\ y = z ) ) ) ) <-> ( D e. Y -> ( { A , B } = { z , D } <-> ( ( A = z /\ B = D ) \/ ( A = D /\ B = z ) ) ) ) ) )
19		preq1 3469	. . . . . . 7 |- ( z = C -> { z , D } = { C , D } )
20	19	eqeq2d 2092	. . . . . 6 |- ( z = C -> ( { A , B } = { z , D } <-> { A , B } = { C , D } ) )
21		eqeq2 2090	. . . . . . . 8 |- ( z = C -> ( A = z <-> A = C ) )
22	21	anbi1d 452	. . . . . . 7 |- ( z = C -> ( ( A = z /\ B = D ) <-> ( A = C /\ B = D ) ) )
23		eqeq2 2090	. . . . . . . 8 |- ( z = C -> ( B = z <-> B = C ) )
24	23	anbi2d 451	. . . . . . 7 |- ( z = C -> ( ( A = D /\ B = z ) <-> ( A = D /\ B = C ) ) )
25	22, 24	orbi12d 739	. . . . . 6 |- ( z = C -> ( ( ( A = z /\ B = D ) \/ ( A = D /\ B = z ) ) <-> ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) ) )
26	20, 25	bibi12d 233	. . . . 5 |- ( z = C -> ( ( { A , B } = { z , D } <-> ( ( A = z /\ B = D ) \/ ( A = D /\ B = z ) ) ) <-> ( { A , B } = { C , D } <-> ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) ) ) )
27	26	imbi2d 228	. . . 4 |- ( z = C -> ( ( D e. Y -> ( { A , B } = { z , D } <-> ( ( A = z /\ B = D ) \/ ( A = D /\ B = z ) ) ) ) <-> ( D e. Y -> ( { A , B } = { C , D } <-> ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) ) ) ) )
28		preq2 3470	. . . . . . 7 |- ( w = D -> { z , w } = { z , D } )
29	28	eqeq2d 2092	. . . . . 6 |- ( w = D -> ( { x , y } = { z , w } <-> { x , y } = { z , D } ) )
30		eqeq2 2090	. . . . . . . 8 |- ( w = D -> ( y = w <-> y = D ) )
31	30	anbi2d 451	. . . . . . 7 |- ( w = D -> ( ( x = z /\ y = w ) <-> ( x = z /\ y = D ) ) )
32		eqeq2 2090	. . . . . . . 8 |- ( w = D -> ( x = w <-> x = D ) )
33	32	anbi1d 452	. . . . . . 7 |- ( w = D -> ( ( x = w /\ y = z ) <-> ( x = D /\ y = z ) ) )
34	31, 33	orbi12d 739	. . . . . 6 |- ( w = D -> ( ( ( x = z /\ y = w ) \/ ( x = w /\ y = z ) ) <-> ( ( x = z /\ y = D ) \/ ( x = D /\ y = z ) ) ) )
35		vex 2604	. . . . . . 7 |- x e. _V
36		vex 2604	. . . . . . 7 |- y e. _V
37		vex 2604	. . . . . . 7 |- z e. _V
38		vex 2604	. . . . . . 7 |- w e. _V
39	35, 36, 37, 38	preq12b 3562	. . . . . 6 |- ( { x , y } = { z , w } <-> ( ( x = z /\ y = w ) \/ ( x = w /\ y = z ) ) )
40	29, 34, 39	vtoclbg 2659	. . . . 5 |- ( D e. Y -> ( { x , y } = { z , D } <-> ( ( x = z /\ y = D ) \/ ( x = D /\ y = z ) ) ) )
41	40	a1i 9	. . . 4 |- ( ( x e. V /\ y e. W /\ z e. X ) -> ( D e. Y -> ( { x , y } = { z , D } <-> ( ( x = z /\ y = D ) \/ ( x = D /\ y = z ) ) ) ) )
42	9, 18, 27, 41	vtocl3ga 2668	. . 3 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( D e. Y -> ( { A , B } = { C , D } <-> ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) ) ) )
43	42	3expa 1138	. 2 |- ( ( ( A e. V /\ B e. W ) /\ C e. X ) -> ( D e. Y -> ( { A , B } = { C , D } <-> ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) ) ) )
44	43	impr 371	1 |- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) ) -> ( { A , B } = { C , D } <-> ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   \/ wo 661   /\ w3a 919   = wceq 1284   e. wcel 1433   {cpr 3399

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-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405

This theorem is referenced by:  prneimg  3566