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Theorem preq12b 3588
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 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)))

Proof of Theorem preq12b
StepHypRef Expression
1 preq12b.1 . . . . . 6 𝐴 ∈ V
21prid1 3522 . . . . 5 𝐴 ∈ {𝐴, 𝐵}
3 eleq2 2146 . . . . 5 ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 ∈ {𝐴, 𝐵} ↔ 𝐴 ∈ {𝐶, 𝐷}))
42, 3mpbii 146 . . . 4 ({𝐴, 𝐵} = {𝐶, 𝐷} → 𝐴 ∈ {𝐶, 𝐷})
51elpr 3443 . . . 4 (𝐴 ∈ {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐴 = 𝐷))
64, 5sylib 120 . . 3 ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 = 𝐶𝐴 = 𝐷))
7 preq1 3493 . . . . . . . 8 (𝐴 = 𝐶 → {𝐴, 𝐵} = {𝐶, 𝐵})
87eqeq1d 2091 . . . . . . 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 3587 . . . . . . 7 ({𝐶, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷)
128, 11syl6bi 161 . . . . . 6 (𝐴 = 𝐶 → ({𝐴, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷))
1312com12 30 . . . . 5 ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 = 𝐶𝐵 = 𝐷))
1413ancld 318 . . . 4 ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 = 𝐶 → (𝐴 = 𝐶𝐵 = 𝐷)))
15 prcom 3492 . . . . . . 7 {𝐶, 𝐷} = {𝐷, 𝐶}
1615eqeq2i 2093 . . . . . 6 ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ {𝐴, 𝐵} = {𝐷, 𝐶})
17 preq1 3493 . . . . . . . . 9 (𝐴 = 𝐷 → {𝐴, 𝐵} = {𝐷, 𝐵})
1817eqeq1d 2091 . . . . . . . 8 (𝐴 = 𝐷 → ({𝐴, 𝐵} = {𝐷, 𝐶} ↔ {𝐷, 𝐵} = {𝐷, 𝐶}))
19 preq12b.3 . . . . . . . . 9 𝐶 ∈ V
209, 19preqr2 3587 . . . . . . . 8 ({𝐷, 𝐵} = {𝐷, 𝐶} → 𝐵 = 𝐶)
2118, 20syl6bi 161 . . . . . . 7 (𝐴 = 𝐷 → ({𝐴, 𝐵} = {𝐷, 𝐶} → 𝐵 = 𝐶))
2221com12 30 . . . . . 6 ({𝐴, 𝐵} = {𝐷, 𝐶} → (𝐴 = 𝐷𝐵 = 𝐶))
2316, 22sylbi 119 . . . . 5 ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 = 𝐷𝐵 = 𝐶))
2423ancld 318 . . . 4 ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 = 𝐷 → (𝐴 = 𝐷𝐵 = 𝐶)))
2514, 24orim12d 733 . . 3 ({𝐴, 𝐵} = {𝐶, 𝐷} → ((𝐴 = 𝐶𝐴 = 𝐷) → ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
266, 25mpd 13 . 2 ({𝐴, 𝐵} = {𝐶, 𝐷} → ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)))
27 preq12 3495 . . 3 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})
28 prcom 3492 . . . . 5 {𝐷, 𝐵} = {𝐵, 𝐷}
2917, 28syl6eq 2131 . . . 4 (𝐴 = 𝐷 → {𝐴, 𝐵} = {𝐵, 𝐷})
30 preq1 3493 . . . 4 (𝐵 = 𝐶 → {𝐵, 𝐷} = {𝐶, 𝐷})
3129, 30sylan9eq 2135 . . 3 ((𝐴 = 𝐷𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐶, 𝐷})
3227, 31jaoi 669 . 2 (((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)) → {𝐴, 𝐵} = {𝐶, 𝐷})
3326, 32impbii 124 1 ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wo 662   = wceq 1285  wcel 1434  Vcvv 2612  {cpr 3423
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-un 2988  df-sn 3428  df-pr 3429
This theorem is referenced by:  prel12  3589  opthpr  3590  preq12bg  3591  preqsn  3593  opeqpr  4044  preleq  4334
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