Theorem preq12b 4382

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

Hypotheses

Ref	Expression
preqr1.a	|- A e. _V
preqr1.b	|- B e. _V
preq12b.c	|- C e. _V
preq12b.d	|- 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		preqr1.a	. . . . . 6 |- A e. _V
2	1	prid1 4297	. . . . 5 |- A e. { A , B }
3		eleq2 2690	. . . . 5 |- ( { A , B } = { C , D } -> ( A e. { A , B } <-> A e. { C , D } ) )
4	2, 3	mpbii 223	. . . 4 |- ( { A , B } = { C , D } -> A e. { C , D } )
5	1	elpr 4198	. . . 4 |- ( A e. { C , D } <-> ( A = C \/ A = D ) )
6	4, 5	sylib 208	. . 3 |- ( { A , B } = { C , D } -> ( A = C \/ A = D ) )
7		preq1 4268	. . . . . . . 8 |- ( A = C -> { A , B } = { C , B } )
8	7	eqeq1d 2624	. . . . . . 7 |- ( A = C -> ( { A , B } = { C , D } <-> { C , B } = { C , D } ) )
9		preqr1.b	. . . . . . . 8 |- B e. _V
10		preq12b.d	. . . . . . . 8 |- D e. _V
11	9, 10	preqr2 4381	. . . . . . 7 |- ( { C , B } = { C , D } -> B = D )
12	8, 11	syl6bi 243	. . . . . 6 |- ( A = C -> ( { A , B } = { C , D } -> B = D ) )
13	12	com12 32	. . . . 5 |- ( { A , B } = { C , D } -> ( A = C -> B = D ) )
14	13	ancld 576	. . . 4 |- ( { A , B } = { C , D } -> ( A = C -> ( A = C /\ B = D ) ) )
15		prcom 4267	. . . . . . 7 |- { C , D } = { D , C }
16	15	eqeq2i 2634	. . . . . 6 |- ( { A , B } = { C , D } <-> { A , B } = { D , C } )
17		preq1 4268	. . . . . . . . 9 |- ( A = D -> { A , B } = { D , B } )
18	17	eqeq1d 2624	. . . . . . . 8 |- ( A = D -> ( { A , B } = { D , C } <-> { D , B } = { D , C } ) )
19		preq12b.c	. . . . . . . . 9 |- C e. _V
20	9, 19	preqr2 4381	. . . . . . . 8 |- ( { D , B } = { D , C } -> B = C )
21	18, 20	syl6bi 243	. . . . . . 7 |- ( A = D -> ( { A , B } = { D , C } -> B = C ) )
22	21	com12 32	. . . . . 6 |- ( { A , B } = { D , C } -> ( A = D -> B = C ) )
23	16, 22	sylbi 207	. . . . 5 |- ( { A , B } = { C , D } -> ( A = D -> B = C ) )
24	23	ancld 576	. . . 4 |- ( { A , B } = { C , D } -> ( A = D -> ( A = D /\ B = C ) ) )
25	14, 24	orim12d 883	. . 3 |- ( { A , B } = { C , D } -> ( ( A = C \/ A = D ) -> ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) ) )
26	6, 25	mpd 15	. 2 |- ( { A , B } = { C , D } -> ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) )
27		preq12 4270	. . 3 |- ( ( A = C /\ B = D ) -> { A , B } = { C , D } )
28		preq12 4270	. . . 4 |- ( ( A = D /\ B = C ) -> { A , B } = { D , C } )
29		prcom 4267	. . . 4 |- { D , C } = { C , D }
30	28, 29	syl6eq 2672	. . 3 |- ( ( A = D /\ B = C ) -> { A , B } = { C , D } )
31	27, 30	jaoi 394	. 2 |- ( ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) -> { A , B } = { C , D } )
32	26, 31	impbii 199	1 |- ( { A , B } = { C , D } <-> ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   {cpr 4179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-sn 4178  df-pr 4180

This theorem is referenced by:  prel12  4383  opthpr  4384  preq12bg  4386  preqsn  4393  preqsnOLD  4394  opeqpr  4968  preleq  8514  axlowdimlem13  25834  upgrwlkdvdelem  26632  altopthsn  32068  sprsymrelfolem2  41743