Theorem preq12b 4128

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

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 3828	. . . . 5 |- A e. {A, B}
3		eleq2 2414	. . . . 5 |- ({A, B} = {C, D} -> (A e. {A, B} <-> A e. {C, D}))
4	2, 3	mpbii 202	. . . 4 |- ({A, B} = {C, D} -> A e. {C, D})
5	1	elpr 3752	. . . 4 |- (A e. {C, D} <-> (A = C \/ A = D))
6	4, 5	sylib 188	. . 3 |- ({A, B} = {C, D} -> (A = C \/ A = D))
7		preq1 3800	. . . . . . . 8 |- (A = C -> {A, B} = {C, B})
8	7	eqeq1d 2361	. . . . . . 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 4126	. . . . . . 7 |- ({C, B} = {C, D} -> B = D)
12	8, 11	syl6bi 219	. . . . . 6 |- (A = C -> ({A, B} = {C, D} -> B = D))
13	12	com12 27	. . . . 5 |- ({A, B} = {C, D} -> (A = C -> B = D))
14	13	ancld 536	. . . 4 |- ({A, B} = {C, D} -> (A = C -> (A = C /\ B = D)))
15		prcom 3799	. . . . . . 7 |- {C, D} = {D, C}
16	15	eqeq2i 2363	. . . . . 6 |- ({A, B} = {C, D} <-> {A, B} = {D, C})
17		preq1 3800	. . . . . . . . 9 |- (A = D -> {A, B} = {D, B})
18	17	eqeq1d 2361	. . . . . . . 8 |- (A = D -> ({A, B} = {D, C} <-> {D, B} = {D, C}))
19		preq12b.3	. . . . . . . . 9 |- C e. _V
20	9, 19	preqr2 4126	. . . . . . . 8 |- ({D, B} = {D, C} -> B = C)
21	18, 20	syl6bi 219	. . . . . . 7 |- (A = D -> ({A, B} = {D, C} -> B = C))
22	21	com12 27	. . . . . 6 |- ({A, B} = {D, C} -> (A = D -> B = C))
23	16, 22	sylbi 187	. . . . 5 |- ({A, B} = {C, D} -> (A = D -> B = C))
24	23	ancld 536	. . . 4 |- ({A, B} = {C, D} -> (A = D -> (A = D /\ B = C)))
25	14, 24	orim12d 811	. . 3 |- ({A, B} = {C, D} -> ((A = C \/ A = D) -> ((A = C /\ B = D) \/ (A = D /\ B = C))))
26	6, 25	mpd 14	. 2 |- ({A, B} = {C, D} -> ((A = C /\ B = D) \/ (A = D /\ B = C)))
27		preq12 3802	. . 3 |- ((A = C /\ B = D) -> {A, B} = {C, D})
28		prcom 3799	. . . . 5 |- {D, B} = {B, D}
29	17, 28	syl6eq 2401	. . . 4 |- (A = D -> {A, B} = {B, D})
30		preq1 3800	. . . 4 |- (B = C -> {B, D} = {C, D})
31	29, 30	sylan9eq 2405	. . 3 |- ((A = D /\ B = C) -> {A, B} = {C, D})
32	27, 31	jaoi 368	. 2 |- (((A = C /\ B = D) \/ (A = D /\ B = C)) -> {A, B} = {C, D})
33	26, 32	impbii 180	1 |- ({A, B} = {C, D} <-> ((A = C /\ B = D) \/ (A = D /\ B = C)))

Syntax hints:   -> wi 4   <-> wb 176   \/ wo 357   /\ wa 358   = wceq 1642   e. wcel 1710  _Vcvv 2860  {cpr 3739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215  df-sn 3742  df-pr 3743

This theorem is referenced by:  preq12bg  4129