Theorem preq12b 2487

Description: Equality relationship for two unordered pairs.

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	pri1 2454	. . . . 5 |- A e. {A, B}
3		eleq2 1538	. . . . 5 |- ({A, B} = {C, D} -> (A e. {A, B} <-> A e. {C, D}))
4	2, 3	mpbii 193	. . . 4 |- ({A, B} = {C, D} -> A e. {C, D})
5	1	elpr 2428	. . . 4 |- (A e. {C, D} <-> (A = C \/ A = D))
6	4, 5	sylib 198	. . 3 |- ({A, B} = {C, D} -> (A = C \/ A = D))
7		preq1 2452	. . . . . . . 8 |- (A = C -> {A, B} = {C, B})
8	7	eqeq1d 1486	. . . . . . 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 2486	. . . . . . 7 |- ({C, B} = {C, D} -> B = D)
12	8, 11	syl6bi 214	. . . . . 6 |- (A = C -> ({A, B} = {C, D} -> B = D))
13	12	com12 11	. . . . 5 |- ({A, B} = {C, D} -> (A = C -> B = D))
14	13	ancld 298	. . . 4 |- ({A, B} = {C, D} -> (A = C -> (A = C /\ B = D)))
15		prcom 2451	. . . . . . 7 |- {C, D} = {D, C}
16	15	eqeq2i 1488	. . . . . 6 |- ({A, B} = {C, D} <-> {A, B} = {D, C})
17		preq1 2452	. . . . . . . . 9 |- (A = D -> {A, B} = {D, B})
18	17	eqeq1d 1486	. . . . . . . 8 |- (A = D -> ({A, B} = {D, C} <-> {D, B} = {D, C}))
19		preq12b.3	. . . . . . . . 9 |- C e. V
20	9, 19	preqr2 2486	. . . . . . . 8 |- ({D, B} = {D, C} -> B = C)
21	18, 20	syl6bi 214	. . . . . . 7 |- (A = D -> ({A, B} = {D, C} -> B = C))
22	21	com12 11	. . . . . 6 |- ({A, B} = {D, C} -> (A = D -> B = C))
23	16, 22	sylbi 199	. . . . 5 |- ({A, B} = {C, D} -> (A = D -> B = C))
24	23	ancld 298	. . . 4 |- ({A, B} = {C, D} -> (A = D -> (A = D /\ B = C)))
25	14, 24	orim12d 567	. . 3 |- ({A, B} = {C, D} -> ((A = C \/ A = D) -> ((A = C /\ B = D) \/ (A = D /\ B = C))))
26	6, 25	mpd 26	. 2 |- ({A, B} = {C, D} -> ((A = C /\ B = D) \/ (A = D /\ B = C)))
27		preq2 2453	. . . 4 |- (B = D -> {C, B} = {C, D})
28	7, 27	sylan9eq 1530	. . 3 |- ((A = C /\ B = D) -> {A, B} = {C, D})
29		prcom 2451	. . . . 5 |- {D, B} = {B, D}
30	17, 29	syl6eq 1526	. . . 4 |- (A = D -> {A, B} = {B, D})
31		preq1 2452	. . . 4 |- (B = C -> {B, D} = {C, D})
32	30, 31	sylan9eq 1530	. . 3 |- ((A = D /\ B = C) -> {A, B} = {C, D})
33	28, 32	jaoi 341	. 2 |- (((A = C /\ B = D) \/ (A = D /\ B = C)) -> {A, B} = {C, D})
34	26, 33	impbi 157	1 |- ({A, B} = {C, D} <-> ((A = C /\ B = D) \/ (A = D /\ B = C)))

Syntax hints:   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 958   e. wcel 960  Vcvv 1814  {cpr 2414

This theorem is referenced by:  prel12 2488  opthpr 2489  preqsn 2490  opeqpr 2809  preleq 4612

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-un 2053  df-sn 2416  df-pr 2417

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