Theorem preq12b 3785

Description: Equality relationship for two unordered pairs. (Contributed by NM, 17-Oct-1996.)

Hypotheses

Ref	Expression
preq12b.1	|- A e. _V
preq12b.2	|- B e. _V
preq12b.3	|- C e. _V
preq12b.4	|- D e. _V

Assertion

Ref	Expression
preq12b	|- ( { A , B } = { C , D } <-> ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) )

Proof of Theorem preq12b

Step	Hyp	Ref	Expression
1		preq12b.1	. . . . . 6 |- A e. _V
2	1	prid1 3713	. . . . 5 |- A e. { A , B }
3		eleq2 2253	. . . . 5 |- ( { A , B } = { C , D } -> ( A e. { A , B } <-> A e. { C , D } ) )
4	2, 3	mpbii 148	. . . 4 |- ( { A , B } = { C , D } -> A e. { C , D } )
5	1	elpr 3628	. . . 4 |- ( A e. { C , D } <-> ( A = C \/ A = D ) )
6	4, 5	sylib 122	. . 3 |- ( { A , B } = { C , D } -> ( A = C \/ A = D ) )
7		preq1 3684	. . . . . . . 8 |- ( A = C -> { A , B } = { C , B } )
8	7	eqeq1d 2198	. . . . . . 7 |- ( A = C -> ( { A , B } = { C , D } <-> { C , B } = { C , D } ) )
9		preq12b.2	. . . . . . . 8 |- B e. _V
10		preq12b.4	. . . . . . . 8 |- D e. _V
11	9, 10	preqr2 3784	. . . . . . 7 |- ( { C , B } = { C , D } -> B = D )
12	8, 11	biimtrdi 163	. . . . . 6 |- ( A = C -> ( { A , B } = { C , D } -> B = D ) )
13	12	com12 30	. . . . 5 |- ( { A , B } = { C , D } -> ( A = C -> B = D ) )
14	13	ancld 325	. . . 4 |- ( { A , B } = { C , D } -> ( A = C -> ( A = C /\ B = D ) ) )
15		prcom 3683	. . . . . . 7 |- { C , D } = { D , C }
16	15	eqeq2i 2200	. . . . . 6 |- ( { A , B } = { C , D } <-> { A , B } = { D , C } )
17		preq1 3684	. . . . . . . . 9 |- ( A = D -> { A , B } = { D , B } )
18	17	eqeq1d 2198	. . . . . . . 8 |- ( A = D -> ( { A , B } = { D , C } <-> { D , B } = { D , C } ) )
19		preq12b.3	. . . . . . . . 9 |- C e. _V
20	9, 19	preqr2 3784	. . . . . . . 8 |- ( { D , B } = { D , C } -> B = C )
21	18, 20	biimtrdi 163	. . . . . . 7 |- ( A = D -> ( { A , B } = { D , C } -> B = C ) )
22	21	com12 30	. . . . . 6 |- ( { A , B } = { D , C } -> ( A = D -> B = C ) )
23	16, 22	sylbi 121	. . . . 5 |- ( { A , B } = { C , D } -> ( A = D -> B = C ) )
24	23	ancld 325	. . . 4 |- ( { A , B } = { C , D } -> ( A = D -> ( A = D /\ B = C ) ) )
25	14, 24	orim12d 787	. . 3 |- ( { A , B } = { C , D } -> ( ( A = C \/ A = D ) -> ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) ) )
26	6, 25	mpd 13	. 2 |- ( { A , B } = { C , D } -> ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) )
27		preq12 3686	. . 3 |- ( ( A = C /\ B = D ) -> { A , B } = { C , D } )
28		prcom 3683	. . . . 5 |- { D , B } = { B , D }
29	17, 28	eqtrdi 2238	. . . 4 |- ( A = D -> { A , B } = { B , D } )
30		preq1 3684	. . . 4 |- ( B = C -> { B , D } = { C , D } )
31	29, 30	sylan9eq 2242	. . 3 |- ( ( A = D /\ B = C ) -> { A , B } = { C , D } )
32	27, 31	jaoi 717	. 2 |- ( ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) -> { A , B } = { C , D } )
33	26, 32	impbii 126	1 |- ( { A , B } = { C , D } <-> ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   = wceq 1364   e. wcel 2160   _Vcvv 2752   {cpr 3608

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614

This theorem is referenced by:  prel12  3786  opthpr  3787  preq12bg  3788  preqsn  3790  opeqpr  4271  preleq  4572