Theorem preq12bg 3804

Description: Closed form of preq12b 3801. (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 3700	. . . . . . 7 |- ( x = A -> { x , y } = { A , y } )
2	1	eqeq1d 2205	. . . . . 6 |- ( x = A -> ( { x , y } = { z , D } <-> { A , y } = { z , D } ) )
3		eqeq1 2203	. . . . . . . 8 |- ( x = A -> ( x = z <-> A = z ) )
4	3	anbi1d 465	. . . . . . 7 |- ( x = A -> ( ( x = z /\ y = D ) <-> ( A = z /\ y = D ) ) )
5		eqeq1 2203	. . . . . . . 8 |- ( x = A -> ( x = D <-> A = D ) )
6	5	anbi1d 465	. . . . . . 7 |- ( x = A -> ( ( x = D /\ y = z ) <-> ( A = D /\ y = z ) ) )
7	4, 6	orbi12d 794	. . . . . 6 |- ( x = A -> ( ( ( x = z /\ y = D ) \/ ( x = D /\ y = z ) ) <-> ( ( A = z /\ y = D ) \/ ( A = D /\ y = z ) ) ) )
8	2, 7	bibi12d 235	. . . . 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 230	. . . 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 3701	. . . . . . 7 |- ( y = B -> { A , y } = { A , B } )
11	10	eqeq1d 2205	. . . . . 6 |- ( y = B -> ( { A , y } = { z , D } <-> { A , B } = { z , D } ) )
12		eqeq1 2203	. . . . . . . 8 |- ( y = B -> ( y = D <-> B = D ) )
13	12	anbi2d 464	. . . . . . 7 |- ( y = B -> ( ( A = z /\ y = D ) <-> ( A = z /\ B = D ) ) )
14		eqeq1 2203	. . . . . . . 8 |- ( y = B -> ( y = z <-> B = z ) )
15	14	anbi2d 464	. . . . . . 7 |- ( y = B -> ( ( A = D /\ y = z ) <-> ( A = D /\ B = z ) ) )
16	13, 15	orbi12d 794	. . . . . 6 |- ( y = B -> ( ( ( A = z /\ y = D ) \/ ( A = D /\ y = z ) ) <-> ( ( A = z /\ B = D ) \/ ( A = D /\ B = z ) ) ) )
17	11, 16	bibi12d 235	. . . . 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 230	. . . 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 3700	. . . . . . 7 |- ( z = C -> { z , D } = { C , D } )
20	19	eqeq2d 2208	. . . . . 6 |- ( z = C -> ( { A , B } = { z , D } <-> { A , B } = { C , D } ) )
21		eqeq2 2206	. . . . . . . 8 |- ( z = C -> ( A = z <-> A = C ) )
22	21	anbi1d 465	. . . . . . 7 |- ( z = C -> ( ( A = z /\ B = D ) <-> ( A = C /\ B = D ) ) )
23		eqeq2 2206	. . . . . . . 8 |- ( z = C -> ( B = z <-> B = C ) )
24	23	anbi2d 464	. . . . . . 7 |- ( z = C -> ( ( A = D /\ B = z ) <-> ( A = D /\ B = C ) ) )
25	22, 24	orbi12d 794	. . . . . 6 |- ( z = C -> ( ( ( A = z /\ B = D ) \/ ( A = D /\ B = z ) ) <-> ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) ) )
26	20, 25	bibi12d 235	. . . . 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 230	. . . 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 3701	. . . . . . 7 |- ( w = D -> { z , w } = { z , D } )
29	28	eqeq2d 2208	. . . . . 6 |- ( w = D -> ( { x , y } = { z , w } <-> { x , y } = { z , D } ) )
30		eqeq2 2206	. . . . . . . 8 |- ( w = D -> ( y = w <-> y = D ) )
31	30	anbi2d 464	. . . . . . 7 |- ( w = D -> ( ( x = z /\ y = w ) <-> ( x = z /\ y = D ) ) )
32		eqeq2 2206	. . . . . . . 8 |- ( w = D -> ( x = w <-> x = D ) )
33	32	anbi1d 465	. . . . . . 7 |- ( w = D -> ( ( x = w /\ y = z ) <-> ( x = D /\ y = z ) ) )
34	31, 33	orbi12d 794	. . . . . 6 |- ( w = D -> ( ( ( x = z /\ y = w ) \/ ( x = w /\ y = z ) ) <-> ( ( x = z /\ y = D ) \/ ( x = D /\ y = z ) ) ) )
35		vex 2766	. . . . . . 7 |- x e. _V
36		vex 2766	. . . . . . 7 |- y e. _V
37		vex 2766	. . . . . . 7 |- z e. _V
38		vex 2766	. . . . . . 7 |- w e. _V
39	35, 36, 37, 38	preq12b 3801	. . . . . 6 |- ( { x , y } = { z , w } <-> ( ( x = z /\ y = w ) \/ ( x = w /\ y = z ) ) )
40	29, 34, 39	vtoclbg 2825	. . . . 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 2834	. . 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 1205	. 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 379	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 104   <-> wb 105   \/ wo 709   /\ w3a 980   = wceq 1364   e. wcel 2167   {cpr 3624

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 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-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-sn 3629  df-pr 3630

This theorem is referenced by:  prneimg  3805  pythagtriplem2  12460  pythagtrip  12477