Theorem prel12 2475

Description: Equality of 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
prel12	|- (-. A = B -> ({A, B} = {C, D} <-> (A e. {C, D} /\ B e. {C, D})))

Proof of Theorem prel12

Step	Hyp	Ref	Expression
1		preq12b.1	. . . . 5 |- A e. V
2	1	pri1 2441	. . . 4 |- A e. {A, B}
3		eleq2 1527	. . . 4 |- ({A, B} = {C, D} -> (A e. {A, B} <-> A e. {C, D}))
4	2, 3	mpbii 193	. . 3 |- ({A, B} = {C, D} -> A e. {C, D})
5		preq12b.2	. . . . 5 |- B e. V
6	5	pri2 2442	. . . 4 |- B e. {A, B}
7		eleq2 1527	. . . 4 |- ({A, B} = {C, D} -> (B e. {A, B} <-> B e. {C, D}))
8	6, 7	mpbii 193	. . 3 |- ({A, B} = {C, D} -> B e. {C, D})
9	4, 8	jca 288	. 2 |- ({A, B} = {C, D} -> (A e. {C, D} /\ B e. {C, D}))
10		eqeq2 1476	. . . . . . . . . . . 12 |- (B = D -> (A = B <-> A = D))
11	10	negbid 609	. . . . . . . . . . 11 |- (B = D -> (-. A = B <-> -. A = D))
12		orel2 252	. . . . . . . . . . 11 |- (-. A = D -> ((A = C \/ A = D) -> A = C))
13	11, 12	syl6bi 214	. . . . . . . . . 10 |- (B = D -> (-. A = B -> ((A = C \/ A = D) -> A = C)))
14	13	com3l 34	. . . . . . . . 9 |- (-. A = B -> ((A = C \/ A = D) -> (B = D -> A = C)))
15	14	imp 350	. . . . . . . 8 |- ((-. A = B /\ (A = C \/ A = D)) -> (B = D -> A = C))
16	15	ancrd 299	. . . . . . 7 |- ((-. A = B /\ (A = C \/ A = D)) -> (B = D -> (A = C /\ B = D)))
17		eqeq2 1476	. . . . . . . . . . . 12 |- (B = C -> (A = B <-> A = C))
18	17	negbid 609	. . . . . . . . . . 11 |- (B = C -> (-. A = B <-> -. A = C))
19		orel1 251	. . . . . . . . . . 11 |- (-. A = C -> ((A = C \/ A = D) -> A = D))
20	18, 19	syl6bi 214	. . . . . . . . . 10 |- (B = C -> (-. A = B -> ((A = C \/ A = D) -> A = D)))
21	20	com3l 34	. . . . . . . . 9 |- (-. A = B -> ((A = C \/ A = D) -> (B = C -> A = D)))
22	21	imp 350	. . . . . . . 8 |- ((-. A = B /\ (A = C \/ A = D)) -> (B = C -> A = D))
23	22	ancrd 299	. . . . . . 7 |- ((-. A = B /\ (A = C \/ A = D)) -> (B = C -> (A = D /\ B = C)))
24	16, 23	orim12d 563	. . . . . 6 |- ((-. A = B /\ (A = C \/ A = D)) -> ((B = D \/ B = C) -> ((A = C /\ B = D) \/ (A = D /\ B = C))))
25	5	elpr 2414	. . . . . . 7 |- (B e. {C, D} <-> (B = C \/ B = D))
26		orcom 246	. . . . . . 7 |- ((B = C \/ B = D) <-> (B = D \/ B = C))
27	25, 26	bitr 173	. . . . . 6 |- (B e. {C, D} <-> (B = D \/ B = C))
28		preq12b.3	. . . . . . 7 |- C e. V
29		preq12b.4	. . . . . . 7 |- D e. V
30	1, 5, 28, 29	preq12b 2474	. . . . . 6 |- ({A, B} = {C, D} <-> ((A = C /\ B = D) \/ (A = D /\ B = C)))
31	24, 27, 30	3imtr4g 551	. . . . 5 |- ((-. A = B /\ (A = C \/ A = D)) -> (B e. {C, D} -> {A, B} = {C, D}))
32	31	ex 373	. . . 4 |- (-. A = B -> ((A = C \/ A = D) -> (B e. {C, D} -> {A, B} = {C, D})))
33	1	elpr 2414	. . . 4 |- (A e. {C, D} <-> (A = C \/ A = D))
34	32, 33	syl5ib 206	. . 3 |- (-. A = B -> (A e. {C, D} -> (B e. {C, D} -> {A, B} = {C, D})))
35	34	imp3a 361	. 2 |- (-. A = B -> ((A e. {C, D} /\ B e. {C, D}) -> {A, B} = {C, D}))
36	9, 35	impbid2 516	1 |- (-. A = B -> ({A, B} = {C, D} <-> (A e. {C, D} /\ B e. {C, D})))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 953   e. wcel 955  Vcvv 1802  {cpr 2400

This theorem is referenced by:  aceq6b 4714

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-un 2040  df-sn 2402  df-pr 2403

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