Theorem preq12b 3154

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	prid1 3106	. . . . 5 |- A e. {A, B}
3		eleq2 1958	. . . . 5 |- ({A, B} = {C, D} -> (A e. {A, B} <-> A e. {C, D}))
4	2, 3	mpbii 210	. . . 4 |- ({A, B} = {C, D} -> A e. {C, D})
5	1	elpr 3061	. . . 4 |- (A e. {C, D} <-> (A = C \/ A = D))
6	4, 5	sylib 215	. . 3 |- ({A, B} = {C, D} -> (A = C \/ A = D))
7		preq1 3098	. . . . . . . 8 |- (A = C -> {A, B} = {C, B})
8	7	eqeq1d 1892	. . . . . . 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 3153	. . . . . . 7 |- ({C, B} = {C, D} -> B = D)
12	8, 11	syl6bi 231	. . . . . 6 |- (A = C -> ({A, B} = {C, D} -> B = D))
13	12	com12 14	. . . . 5 |- ({A, B} = {C, D} -> (A = C -> B = D))
14	13	ancld 322	. . . 4 |- ({A, B} = {C, D} -> (A = C -> (A = C /\ B = D)))
15		prcom 3097	. . . . . . 7 |- {C, D} = {D, C}
16	15	eqeq2i 1894	. . . . . 6 |- ({A, B} = {C, D} <-> {A, B} = {D, C})
17		preq1 3098	. . . . . . . . 9 |- (A = D -> {A, B} = {D, B})
18	17	eqeq1d 1892	. . . . . . . 8 |- (A = D -> ({A, B} = {D, C} <-> {D, B} = {D, C}))
19		preq12b.3	. . . . . . . . 9 |- C e. _V
20	9, 19	preqr2 3153	. . . . . . . 8 |- ({D, B} = {D, C} -> B = C)
21	18, 20	syl6bi 231	. . . . . . 7 |- (A = D -> ({A, B} = {D, C} -> B = C))
22	21	com12 14	. . . . . 6 |- ({A, B} = {D, C} -> (A = D -> B = C))
23	16, 22	sylbi 216	. . . . 5 |- ({A, B} = {C, D} -> (A = D -> B = C))
24	23	ancld 322	. . . 4 |- ({A, B} = {C, D} -> (A = D -> (A = D /\ B = C)))
25	14, 24	orim12d 624	. . 3 |- ({A, B} = {C, D} -> ((A = C \/ A = D) -> ((A = C /\ B = D) \/ (A = D /\ B = C))))
26	6, 25	mpd 29	. 2 |- ({A, B} = {C, D} -> ((A = C /\ B = D) \/ (A = D /\ B = C)))
27		preq2 3099	. . . 4 |- (B = D -> {C, B} = {C, D})
28	7, 27	sylan9eq 1948	. . 3 |- ((A = C /\ B = D) -> {A, B} = {C, D})
29		prcom 3097	. . . . 5 |- {D, B} = {B, D}
30	17, 29	syl6eq 1944	. . . 4 |- (A = D -> {A, B} = {B, D})
31		preq1 3098	. . . 4 |- (B = C -> {B, D} = {C, D})
32	30, 31	sylan9eq 1948	. . 3 |- ((A = D /\ B = C) -> {A, B} = {C, D})
33	28, 32	jaoi 368	. 2 |- (((A = C /\ B = D) \/ (A = D /\ B = C)) -> {A, B} = {C, D})
34	26, 33	impbii 174	1 |- ({A, B} = {C, D} <-> ((A = C /\ B = D) \/ (A = D /\ B = C)))

Syntax hints:   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292  {cpr 3045

This theorem is referenced by:  prel12 3155  opthpr 3156  preqsn 3157  opeqpr 3550  preleq 5708  altopth1sn 14090  cbcpcp 14504

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-un 2600  df-sn 3049  df-pr 3050

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