Theorem prel12g 4212

Description: Closed form of prel12 4209. (Contributed by AV, 9-Dec-2018.)

Assertion

Ref	Expression
prel12g	|- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) ) -> ( -. A = B -> ( { A , B } = { C , D } <-> ( A e. { C , D } /\ B e. { C , D } ) ) ) )

Proof of Theorem prel12g

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

Step	Hyp	Ref	Expression
1		eqeq1 2461	. . . . . . 7 |- ( x = A -> ( x = y <-> A = y ) )
2	1	notbid 294	. . . . . 6 |- ( x = A -> ( -. x = y <-> -. A = y ) )
3		preq1 4111	. . . . . . . 8 |- ( x = A -> { x , y } = { A , y } )
4	3	eqeq1d 2459	. . . . . . 7 |- ( x = A -> ( { x , y } = { z , D } <-> { A , y } = { z , D } ) )
5		eleq1 2529	. . . . . . . 8 |- ( x = A -> ( x e. { z , D } <-> A e. { z , D } ) )
6	5	anbi1d 704	. . . . . . 7 |- ( x = A -> ( ( x e. { z , D } /\ y e. { z , D } ) <-> ( A e. { z , D } /\ y e. { z , D } ) ) )
7	4, 6	bibi12d 321	. . . . . 6 |- ( x = A -> ( ( { x , y } = { z , D } <-> ( x e. { z , D } /\ y e. { z , D } ) ) <-> ( { A , y } = { z , D } <-> ( A e. { z , D } /\ y e. { z , D } ) ) ) )
8	2, 7	imbi12d 320	. . . . 5 |- ( x = A -> ( ( -. x = y -> ( { x , y } = { z , D } <-> ( x e. { z , D } /\ y e. { z , D } ) ) ) <-> ( -. A = y -> ( { A , y } = { z , D } <-> ( A e. { z , D } /\ y e. { z , D } ) ) ) ) )
9	8	imbi2d 316	. . . 4 |- ( x = A -> ( ( D e. Y -> ( -. x = y -> ( { x , y } = { z , D } <-> ( x e. { z , D } /\ y e. { z , D } ) ) ) ) <-> ( D e. Y -> ( -. A = y -> ( { A , y } = { z , D } <-> ( A e. { z , D } /\ y e. { z , D } ) ) ) ) ) )
10		eqeq2 2472	. . . . . . 7 |- ( y = B -> ( A = y <-> A = B ) )
11	10	notbid 294	. . . . . 6 |- ( y = B -> ( -. A = y <-> -. A = B ) )
12		preq2 4112	. . . . . . . 8 |- ( y = B -> { A , y } = { A , B } )
13	12	eqeq1d 2459	. . . . . . 7 |- ( y = B -> ( { A , y } = { z , D } <-> { A , B } = { z , D } ) )
14		eleq1 2529	. . . . . . . 8 |- ( y = B -> ( y e. { z , D } <-> B e. { z , D } ) )
15	14	anbi2d 703	. . . . . . 7 |- ( y = B -> ( ( A e. { z , D } /\ y e. { z , D } ) <-> ( A e. { z , D } /\ B e. { z , D } ) ) )
16	13, 15	bibi12d 321	. . . . . 6 |- ( y = B -> ( ( { A , y } = { z , D } <-> ( A e. { z , D } /\ y e. { z , D } ) ) <-> ( { A , B } = { z , D } <-> ( A e. { z , D } /\ B e. { z , D } ) ) ) )
17	11, 16	imbi12d 320	. . . . 5 |- ( y = B -> ( ( -. A = y -> ( { A , y } = { z , D } <-> ( A e. { z , D } /\ y e. { z , D } ) ) ) <-> ( -. A = B -> ( { A , B } = { z , D } <-> ( A e. { z , D } /\ B e. { z , D } ) ) ) ) )
18	17	imbi2d 316	. . . 4 |- ( y = B -> ( ( D e. Y -> ( -. A = y -> ( { A , y } = { z , D } <-> ( A e. { z , D } /\ y e. { z , D } ) ) ) ) <-> ( D e. Y -> ( -. A = B -> ( { A , B } = { z , D } <-> ( A e. { z , D } /\ B e. { z , D } ) ) ) ) ) )
19		preq1 4111	. . . . . . . 8 |- ( z = C -> { z , D } = { C , D } )
20	19	eqeq2d 2471	. . . . . . 7 |- ( z = C -> ( { A , B } = { z , D } <-> { A , B } = { C , D } ) )
21	19	eleq2d 2527	. . . . . . . 8 |- ( z = C -> ( A e. { z , D } <-> A e. { C , D } ) )
22	19	eleq2d 2527	. . . . . . . 8 |- ( z = C -> ( B e. { z , D } <-> B e. { C , D } ) )
23	21, 22	anbi12d 710	. . . . . . 7 |- ( z = C -> ( ( A e. { z , D } /\ B e. { z , D } ) <-> ( A e. { C , D } /\ B e. { C , D } ) ) )
24	20, 23	bibi12d 321	. . . . . 6 |- ( z = C -> ( ( { A , B } = { z , D } <-> ( A e. { z , D } /\ B e. { z , D } ) ) <-> ( { A , B } = { C , D } <-> ( A e. { C , D } /\ B e. { C , D } ) ) ) )
25	24	imbi2d 316	. . . . 5 |- ( z = C -> ( ( -. A = B -> ( { A , B } = { z , D } <-> ( A e. { z , D } /\ B e. { z , D } ) ) ) <-> ( -. A = B -> ( { A , B } = { C , D } <-> ( A e. { C , D } /\ B e. { C , D } ) ) ) ) )
26	25	imbi2d 316	. . . 4 |- ( z = C -> ( ( D e. Y -> ( -. A = B -> ( { A , B } = { z , D } <-> ( A e. { z , D } /\ B e. { z , D } ) ) ) ) <-> ( D e. Y -> ( -. A = B -> ( { A , B } = { C , D } <-> ( A e. { C , D } /\ B e. { C , D } ) ) ) ) ) )
27		preq2 4112	. . . . . . . . 9 |- ( w = D -> { z , w } = { z , D } )
28	27	eqeq2d 2471	. . . . . . . 8 |- ( w = D -> ( { x , y } = { z , w } <-> { x , y } = { z , D } ) )
29	27	eleq2d 2527	. . . . . . . . 9 |- ( w = D -> ( x e. { z , w } <-> x e. { z , D } ) )
30	27	eleq2d 2527	. . . . . . . . 9 |- ( w = D -> ( y e. { z , w } <-> y e. { z , D } ) )
31	29, 30	anbi12d 710	. . . . . . . 8 |- ( w = D -> ( ( x e. { z , w } /\ y e. { z , w } ) <-> ( x e. { z , D } /\ y e. { z , D } ) ) )
32	28, 31	bibi12d 321	. . . . . . 7 |- ( w = D -> ( ( { x , y } = { z , w } <-> ( x e. { z , w } /\ y e. { z , w } ) ) <-> ( { x , y } = { z , D } <-> ( x e. { z , D } /\ y e. { z , D } ) ) ) )
33	32	imbi2d 316	. . . . . 6 |- ( w = D -> ( ( -. x = y -> ( { x , y } = { z , w } <-> ( x e. { z , w } /\ y e. { z , w } ) ) ) <-> ( -. x = y -> ( { x , y } = { z , D } <-> ( x e. { z , D } /\ y e. { z , D } ) ) ) ) )
34		vex 3112	. . . . . . 7 |- x e. _V
35		vex 3112	. . . . . . 7 |- y e. _V
36		vex 3112	. . . . . . 7 |- z e. _V
37		vex 3112	. . . . . . 7 |- w e. _V
38	34, 35, 36, 37	prel12 4209	. . . . . 6 |- ( -. x = y -> ( { x , y } = { z , w } <-> ( x e. { z , w } /\ y e. { z , w } ) ) )
39	33, 38	vtoclg 3167	. . . . 5 |- ( D e. Y -> ( -. x = y -> ( { x , y } = { z , D } <-> ( x e. { z , D } /\ y e. { z , D } ) ) ) )
40	39	a1i 11	. . . 4 |- ( ( x e. V /\ y e. W /\ z e. X ) -> ( D e. Y -> ( -. x = y -> ( { x , y } = { z , D } <-> ( x e. { z , D } /\ y e. { z , D } ) ) ) ) )
41	9, 18, 26, 40	vtocl3ga 3177	. . 3 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( D e. Y -> ( -. A = B -> ( { A , B } = { C , D } <-> ( A e. { C , D } /\ B e. { C , D } ) ) ) ) )
42	41	3expa 1196	. 2 |- ( ( ( A e. V /\ B e. W ) /\ C e. X ) -> ( D e. Y -> ( -. A = B -> ( { A , B } = { C , D } <-> ( A e. { C , D } /\ B e. { C , D } ) ) ) ) )
43	42	impr 619	1 |- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) ) -> ( -. A = B -> ( { A , B } = { C , D } <-> ( A e. { C , D } /\ B e. { C , D } ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1395   e. wcel 1819   {cpr 4034

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-un 3476  df-sn 4033  df-pr 4035

This theorem is referenced by:  hash2prd  12522