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Theorem preq12bg 3585
Description: Closed form of preq12b 3582. (Contributed by Scott Fenton, 28-Mar-2014.)
Assertion
Ref Expression
preq12bg (((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑋𝐷𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))

Proof of Theorem preq12bg
Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 preq1 3487 . . . . . . 7 (𝑥 = 𝐴 → {𝑥, 𝑦} = {𝐴, 𝑦})
21eqeq1d 2091 . . . . . 6 (𝑥 = 𝐴 → ({𝑥, 𝑦} = {𝑧, 𝐷} ↔ {𝐴, 𝑦} = {𝑧, 𝐷}))
3 eqeq1 2089 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥 = 𝑧𝐴 = 𝑧))
43anbi1d 453 . . . . . . 7 (𝑥 = 𝐴 → ((𝑥 = 𝑧𝑦 = 𝐷) ↔ (𝐴 = 𝑧𝑦 = 𝐷)))
5 eqeq1 2089 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥 = 𝐷𝐴 = 𝐷))
65anbi1d 453 . . . . . . 7 (𝑥 = 𝐴 → ((𝑥 = 𝐷𝑦 = 𝑧) ↔ (𝐴 = 𝐷𝑦 = 𝑧)))
74, 6orbi12d 740 . . . . . 6 (𝑥 = 𝐴 → (((𝑥 = 𝑧𝑦 = 𝐷) ∨ (𝑥 = 𝐷𝑦 = 𝑧)) ↔ ((𝐴 = 𝑧𝑦 = 𝐷) ∨ (𝐴 = 𝐷𝑦 = 𝑧))))
82, 7bibi12d 233 . . . . 5 (𝑥 = 𝐴 → (({𝑥, 𝑦} = {𝑧, 𝐷} ↔ ((𝑥 = 𝑧𝑦 = 𝐷) ∨ (𝑥 = 𝐷𝑦 = 𝑧))) ↔ ({𝐴, 𝑦} = {𝑧, 𝐷} ↔ ((𝐴 = 𝑧𝑦 = 𝐷) ∨ (𝐴 = 𝐷𝑦 = 𝑧)))))
98imbi2d 228 . . . 4 (𝑥 = 𝐴 → ((𝐷𝑌 → ({𝑥, 𝑦} = {𝑧, 𝐷} ↔ ((𝑥 = 𝑧𝑦 = 𝐷) ∨ (𝑥 = 𝐷𝑦 = 𝑧)))) ↔ (𝐷𝑌 → ({𝐴, 𝑦} = {𝑧, 𝐷} ↔ ((𝐴 = 𝑧𝑦 = 𝐷) ∨ (𝐴 = 𝐷𝑦 = 𝑧))))))
10 preq2 3488 . . . . . . 7 (𝑦 = 𝐵 → {𝐴, 𝑦} = {𝐴, 𝐵})
1110eqeq1d 2091 . . . . . 6 (𝑦 = 𝐵 → ({𝐴, 𝑦} = {𝑧, 𝐷} ↔ {𝐴, 𝐵} = {𝑧, 𝐷}))
12 eqeq1 2089 . . . . . . . 8 (𝑦 = 𝐵 → (𝑦 = 𝐷𝐵 = 𝐷))
1312anbi2d 452 . . . . . . 7 (𝑦 = 𝐵 → ((𝐴 = 𝑧𝑦 = 𝐷) ↔ (𝐴 = 𝑧𝐵 = 𝐷)))
14 eqeq1 2089 . . . . . . . 8 (𝑦 = 𝐵 → (𝑦 = 𝑧𝐵 = 𝑧))
1514anbi2d 452 . . . . . . 7 (𝑦 = 𝐵 → ((𝐴 = 𝐷𝑦 = 𝑧) ↔ (𝐴 = 𝐷𝐵 = 𝑧)))
1613, 15orbi12d 740 . . . . . 6 (𝑦 = 𝐵 → (((𝐴 = 𝑧𝑦 = 𝐷) ∨ (𝐴 = 𝐷𝑦 = 𝑧)) ↔ ((𝐴 = 𝑧𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝑧))))
1711, 16bibi12d 233 . . . . 5 (𝑦 = 𝐵 → (({𝐴, 𝑦} = {𝑧, 𝐷} ↔ ((𝐴 = 𝑧𝑦 = 𝐷) ∨ (𝐴 = 𝐷𝑦 = 𝑧))) ↔ ({𝐴, 𝐵} = {𝑧, 𝐷} ↔ ((𝐴 = 𝑧𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝑧)))))
1817imbi2d 228 . . . 4 (𝑦 = 𝐵 → ((𝐷𝑌 → ({𝐴, 𝑦} = {𝑧, 𝐷} ↔ ((𝐴 = 𝑧𝑦 = 𝐷) ∨ (𝐴 = 𝐷𝑦 = 𝑧)))) ↔ (𝐷𝑌 → ({𝐴, 𝐵} = {𝑧, 𝐷} ↔ ((𝐴 = 𝑧𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝑧))))))
19 preq1 3487 . . . . . . 7 (𝑧 = 𝐶 → {𝑧, 𝐷} = {𝐶, 𝐷})
2019eqeq2d 2094 . . . . . 6 (𝑧 = 𝐶 → ({𝐴, 𝐵} = {𝑧, 𝐷} ↔ {𝐴, 𝐵} = {𝐶, 𝐷}))
21 eqeq2 2092 . . . . . . . 8 (𝑧 = 𝐶 → (𝐴 = 𝑧𝐴 = 𝐶))
2221anbi1d 453 . . . . . . 7 (𝑧 = 𝐶 → ((𝐴 = 𝑧𝐵 = 𝐷) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
23 eqeq2 2092 . . . . . . . 8 (𝑧 = 𝐶 → (𝐵 = 𝑧𝐵 = 𝐶))
2423anbi2d 452 . . . . . . 7 (𝑧 = 𝐶 → ((𝐴 = 𝐷𝐵 = 𝑧) ↔ (𝐴 = 𝐷𝐵 = 𝐶)))
2522, 24orbi12d 740 . . . . . 6 (𝑧 = 𝐶 → (((𝐴 = 𝑧𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝑧)) ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
2620, 25bibi12d 233 . . . . 5 (𝑧 = 𝐶 → (({𝐴, 𝐵} = {𝑧, 𝐷} ↔ ((𝐴 = 𝑧𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝑧))) ↔ ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)))))
2726imbi2d 228 . . . 4 (𝑧 = 𝐶 → ((𝐷𝑌 → ({𝐴, 𝐵} = {𝑧, 𝐷} ↔ ((𝐴 = 𝑧𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝑧)))) ↔ (𝐷𝑌 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))))
28 preq2 3488 . . . . . . 7 (𝑤 = 𝐷 → {𝑧, 𝑤} = {𝑧, 𝐷})
2928eqeq2d 2094 . . . . . 6 (𝑤 = 𝐷 → ({𝑥, 𝑦} = {𝑧, 𝑤} ↔ {𝑥, 𝑦} = {𝑧, 𝐷}))
30 eqeq2 2092 . . . . . . . 8 (𝑤 = 𝐷 → (𝑦 = 𝑤𝑦 = 𝐷))
3130anbi2d 452 . . . . . . 7 (𝑤 = 𝐷 → ((𝑥 = 𝑧𝑦 = 𝑤) ↔ (𝑥 = 𝑧𝑦 = 𝐷)))
32 eqeq2 2092 . . . . . . . 8 (𝑤 = 𝐷 → (𝑥 = 𝑤𝑥 = 𝐷))
3332anbi1d 453 . . . . . . 7 (𝑤 = 𝐷 → ((𝑥 = 𝑤𝑦 = 𝑧) ↔ (𝑥 = 𝐷𝑦 = 𝑧)))
3431, 33orbi12d 740 . . . . . 6 (𝑤 = 𝐷 → (((𝑥 = 𝑧𝑦 = 𝑤) ∨ (𝑥 = 𝑤𝑦 = 𝑧)) ↔ ((𝑥 = 𝑧𝑦 = 𝐷) ∨ (𝑥 = 𝐷𝑦 = 𝑧))))
35 vex 2612 . . . . . . 7 𝑥 ∈ V
36 vex 2612 . . . . . . 7 𝑦 ∈ V
37 vex 2612 . . . . . . 7 𝑧 ∈ V
38 vex 2612 . . . . . . 7 𝑤 ∈ V
3935, 36, 37, 38preq12b 3582 . . . . . 6 ({𝑥, 𝑦} = {𝑧, 𝑤} ↔ ((𝑥 = 𝑧𝑦 = 𝑤) ∨ (𝑥 = 𝑤𝑦 = 𝑧)))
4029, 34, 39vtoclbg 2667 . . . . 5 (𝐷𝑌 → ({𝑥, 𝑦} = {𝑧, 𝐷} ↔ ((𝑥 = 𝑧𝑦 = 𝐷) ∨ (𝑥 = 𝐷𝑦 = 𝑧))))
4140a1i 9 . . . 4 ((𝑥𝑉𝑦𝑊𝑧𝑋) → (𝐷𝑌 → ({𝑥, 𝑦} = {𝑧, 𝐷} ↔ ((𝑥 = 𝑧𝑦 = 𝐷) ∨ (𝑥 = 𝐷𝑦 = 𝑧)))))
429, 18, 27, 41vtocl3ga 2676 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐷𝑌 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)))))
43423expa 1139 . 2 (((𝐴𝑉𝐵𝑊) ∧ 𝐶𝑋) → (𝐷𝑌 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)))))
4443impr 371 1 (((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑋𝐷𝑌)) → ({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  w3a 920   = wceq 1285  wcel 1434  {cpr 3417
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-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2611  df-un 2986  df-sn 3422  df-pr 3423
This theorem is referenced by:  prneimg  3586
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