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Theorem preq12b 4127
 Description: Equality relationship for two unordered pairs. (Contributed by NM, 17-Oct-1996.)
Hypotheses
Ref Expression
preq12b.1 A V
preq12b.2 B V
preq12b.3 C V
preq12b.4 D V
Assertion
Ref Expression
preq12b ({A, B} = {C, D} ↔ ((A = C B = D) (A = D B = C)))

Proof of Theorem preq12b
StepHypRef Expression
1 preq12b.1 . . . . . 6 A V
21prid1 3827 . . . . 5 A {A, B}
3 eleq2 2414 . . . . 5 ({A, B} = {C, D} → (A {A, B} ↔ A {C, D}))
42, 3mpbii 202 . . . 4 ({A, B} = {C, D} → A {C, D})
51elpr 3751 . . . 4 (A {C, D} ↔ (A = C A = D))
64, 5sylib 188 . . 3 ({A, B} = {C, D} → (A = C A = D))
7 preq1 3799 . . . . . . . 8 (A = C → {A, B} = {C, B})
87eqeq1d 2361 . . . . . . 7 (A = C → ({A, B} = {C, D} ↔ {C, B} = {C, D}))
9 preq12b.2 . . . . . . . 8 B V
10 preq12b.4 . . . . . . . 8 D V
119, 10preqr2 4125 . . . . . . 7 ({C, B} = {C, D} → B = D)
128, 11syl6bi 219 . . . . . 6 (A = C → ({A, B} = {C, D} → B = D))
1312com12 27 . . . . 5 ({A, B} = {C, D} → (A = CB = D))
1413ancld 536 . . . 4 ({A, B} = {C, D} → (A = C → (A = C B = D)))
15 prcom 3798 . . . . . . 7 {C, D} = {D, C}
1615eqeq2i 2363 . . . . . 6 ({A, B} = {C, D} ↔ {A, B} = {D, C})
17 preq1 3799 . . . . . . . . 9 (A = D → {A, B} = {D, B})
1817eqeq1d 2361 . . . . . . . 8 (A = D → ({A, B} = {D, C} ↔ {D, B} = {D, C}))
19 preq12b.3 . . . . . . . . 9 C V
209, 19preqr2 4125 . . . . . . . 8 ({D, B} = {D, C} → B = C)
2118, 20syl6bi 219 . . . . . . 7 (A = D → ({A, B} = {D, C} → B = C))
2221com12 27 . . . . . 6 ({A, B} = {D, C} → (A = DB = C))
2316, 22sylbi 187 . . . . 5 ({A, B} = {C, D} → (A = DB = C))
2423ancld 536 . . . 4 ({A, B} = {C, D} → (A = D → (A = D B = C)))
2514, 24orim12d 811 . . 3 ({A, B} = {C, D} → ((A = C A = D) → ((A = C B = D) (A = D B = C))))
266, 25mpd 14 . 2 ({A, B} = {C, D} → ((A = C B = D) (A = D B = C)))
27 preq12 3801 . . 3 ((A = C B = D) → {A, B} = {C, D})
28 prcom 3798 . . . . 5 {D, B} = {B, D}
2917, 28syl6eq 2401 . . . 4 (A = D → {A, B} = {B, D})
30 preq1 3799 . . . 4 (B = C → {B, D} = {C, D})
3129, 30sylan9eq 2405 . . 3 ((A = D B = C) → {A, B} = {C, D})
3227, 31jaoi 368 . 2 (((A = C B = D) (A = D B = C)) → {A, B} = {C, D})
3326, 32impbii 180 1 ({A, B} = {C, D} ↔ ((A = C B = D) (A = D B = C)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Vcvv 2859  {cpr 3738 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2478  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741  df-pr 3742 This theorem is referenced by:  preq12bg  4128
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