Theorem prel12 1875

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	. . . . . 6 |- A e. V
2	1	pri1 1841	. . . . 5 |- A e. {A, B}
3		eleq2 1150	. . . . 5 |- ({A, B} = {C, D} -> (A e. {A, B} <-> A e. {C, D}))
4	2, 3	mpbii 168	. . . 4 |- ({A, B} = {C, D} -> A e. {C, D})
5		preq12b.2	. . . . . 6 |- B e. V
6	5	pri2 1842	. . . . 5 |- B e. {A, B}
7		eleq2 1150	. . . . 5 |- ({A, B} = {C, D} -> (B e. {A, B} <-> B e. {C, D}))
8	6, 7	mpbii 168	. . . 4 |- ({A, B} = {C, D} -> B e. {C, D})
9	4, 8	jca 236	. . 3 |- ({A, B} = {C, D} -> (A e. {C, D} /\ B e. {C, D}))
10	9	a1i 7	. 2 |- (-. A = B -> ({A, B} = {C, D} -> (A e. {C, D} /\ B e. {C, D})))
11		cleq2 1110	. . . . . . . . . . . 12 |- (B = D -> (A = B <-> A = D))
12	11	negbid 463	. . . . . . . . . . 11 |- (B = D -> (-. A = B <-> -. A = D))
13		orel2 213	. . . . . . . . . . 11 |- (-. A = D -> ((A = C \/ A = D) -> A = C))
14	12, 13	syl6bi 187	. . . . . . . . . 10 |- (B = D -> (-. A = B -> ((A = C \/ A = D) -> A = C)))
15	14	com3l 34	. . . . . . . . 9 |- (-. A = B -> ((A = C \/ A = D) -> (B = D -> A = C)))
16	15	imp 277	. . . . . . . 8 |- ((-. A = B /\ (A = C \/ A = D)) -> (B = D -> A = C))
17	16	ancrd 247	. . . . . . 7 |- ((-. A = B /\ (A = C \/ A = D)) -> (B = D -> (A = C /\ B = D)))
18		cleq2 1110	. . . . . . . . . . . 12 |- (B = C -> (A = B <-> A = C))
19	18	negbid 463	. . . . . . . . . . 11 |- (B = C -> (-. A = B <-> -. A = C))
20		orel1 212	. . . . . . . . . . 11 |- (-. A = C -> ((A = C \/ A = D) -> A = D))
21	19, 20	syl6bi 187	. . . . . . . . . 10 |- (B = C -> (-. A = B -> ((A = C \/ A = D) -> A = D)))
22	21	com3l 34	. . . . . . . . 9 |- (-. A = B -> ((A = C \/ A = D) -> (B = C -> A = D)))
23	22	imp 277	. . . . . . . 8 |- ((-. A = B /\ (A = C \/ A = D)) -> (B = C -> A = D))
24	23	ancrd 247	. . . . . . 7 |- ((-. A = B /\ (A = C \/ A = D)) -> (B = C -> (A = D /\ B = C)))
25	17, 24	orim12d 436	. . . . . 6 |- ((-. A = B /\ (A = C \/ A = D)) -> ((B = D \/ B = C) -> ((A = C /\ B = D) \/ (A = D /\ B = C))))
26	5	elpr 1823	. . . . . . 7 |- (B e. {C, D} <-> (B = C \/ B = D))
27		orcom 209	. . . . . . 7 |- ((B = C \/ B = D) <-> (B = D \/ B = C))
28	26, 27	bitr 151	. . . . . 6 |- (B e. {C, D} <-> (B = D \/ B = C))
29		preq12b.3	. . . . . . 7 |- C e. V
30		preq12b.4	. . . . . . 7 |- D e. V
31	1, 5, 29, 30	preq12b 1874	. . . . . 6 |- ({A, B} = {C, D} <-> ((A = C /\ B = D) \/ (A = D /\ B = C)))
32	25, 28, 31	3imtr4g 426	. . . . 5 |- ((-. A = B /\ (A = C \/ A = D)) -> (B e. {C, D} -> {A, B} = {C, D}))
33	32	exp 291	. . . 4 |- (-. A = B -> ((A = C \/ A = D) -> (B e. {C, D} -> {A, B} = {C, D})))
34	1	elpr 1823	. . . 4 |- (A e. {C, D} <-> (A = C \/ A = D))
35	33, 34	syl5ib 181	. . 3 |- (-. A = B -> (A e. {C, D} -> (B e. {C, D} -> {A, B} = {C, D})))
36	35	imp3a 279	. 2 |- (-. A = B -> ((A e. {C, D} /\ B e. {C, D}) -> {A, B} = {C, D}))
37	10, 36	impbid 397	1 |- (-. A = B -> ({A, B} = {C, D} <-> (A e. {C, D} /\ B e. {C, D})))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   \/ wo 195   /\ wa 196   = wceq 1091   e. wcel 1092  Vcvv 1348  {cpr 1809

This theorem is referenced by:  aceq6b 3565

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-sn 1811  df-pr 1812

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