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Theorem preq12bg 4847
Description: Closed form of preq12b 4844. (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 4730 . . . . . . 7 (𝑥 = 𝐴 → {𝑥, 𝑦} = {𝐴, 𝑦})
21eqeq1d 2733 . . . . . 6 (𝑥 = 𝐴 → ({𝑥, 𝑦} = {𝑧, 𝐷} ↔ {𝐴, 𝑦} = {𝑧, 𝐷}))
3 eqeq1 2735 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥 = 𝑧𝐴 = 𝑧))
43anbi1d 630 . . . . . . 7 (𝑥 = 𝐴 → ((𝑥 = 𝑧𝑦 = 𝐷) ↔ (𝐴 = 𝑧𝑦 = 𝐷)))
5 eqeq1 2735 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥 = 𝐷𝐴 = 𝐷))
65anbi1d 630 . . . . . . 7 (𝑥 = 𝐴 → ((𝑥 = 𝐷𝑦 = 𝑧) ↔ (𝐴 = 𝐷𝑦 = 𝑧)))
74, 6orbi12d 917 . . . . . 6 (𝑥 = 𝐴 → (((𝑥 = 𝑧𝑦 = 𝐷) ∨ (𝑥 = 𝐷𝑦 = 𝑧)) ↔ ((𝐴 = 𝑧𝑦 = 𝐷) ∨ (𝐴 = 𝐷𝑦 = 𝑧))))
82, 7bibi12d 345 . . . . 5 (𝑥 = 𝐴 → (({𝑥, 𝑦} = {𝑧, 𝐷} ↔ ((𝑥 = 𝑧𝑦 = 𝐷) ∨ (𝑥 = 𝐷𝑦 = 𝑧))) ↔ ({𝐴, 𝑦} = {𝑧, 𝐷} ↔ ((𝐴 = 𝑧𝑦 = 𝐷) ∨ (𝐴 = 𝐷𝑦 = 𝑧)))))
98imbi2d 340 . . . 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 4731 . . . . . . 7 (𝑦 = 𝐵 → {𝐴, 𝑦} = {𝐴, 𝐵})
1110eqeq1d 2733 . . . . . 6 (𝑦 = 𝐵 → ({𝐴, 𝑦} = {𝑧, 𝐷} ↔ {𝐴, 𝐵} = {𝑧, 𝐷}))
12 eqeq1 2735 . . . . . . . 8 (𝑦 = 𝐵 → (𝑦 = 𝐷𝐵 = 𝐷))
1312anbi2d 629 . . . . . . 7 (𝑦 = 𝐵 → ((𝐴 = 𝑧𝑦 = 𝐷) ↔ (𝐴 = 𝑧𝐵 = 𝐷)))
14 eqeq1 2735 . . . . . . . 8 (𝑦 = 𝐵 → (𝑦 = 𝑧𝐵 = 𝑧))
1514anbi2d 629 . . . . . . 7 (𝑦 = 𝐵 → ((𝐴 = 𝐷𝑦 = 𝑧) ↔ (𝐴 = 𝐷𝐵 = 𝑧)))
1613, 15orbi12d 917 . . . . . 6 (𝑦 = 𝐵 → (((𝐴 = 𝑧𝑦 = 𝐷) ∨ (𝐴 = 𝐷𝑦 = 𝑧)) ↔ ((𝐴 = 𝑧𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝑧))))
1711, 16bibi12d 345 . . . . 5 (𝑦 = 𝐵 → (({𝐴, 𝑦} = {𝑧, 𝐷} ↔ ((𝐴 = 𝑧𝑦 = 𝐷) ∨ (𝐴 = 𝐷𝑦 = 𝑧))) ↔ ({𝐴, 𝐵} = {𝑧, 𝐷} ↔ ((𝐴 = 𝑧𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝑧)))))
1817imbi2d 340 . . . 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 4730 . . . . . . 7 (𝑧 = 𝐶 → {𝑧, 𝐷} = {𝐶, 𝐷})
2019eqeq2d 2742 . . . . . 6 (𝑧 = 𝐶 → ({𝐴, 𝐵} = {𝑧, 𝐷} ↔ {𝐴, 𝐵} = {𝐶, 𝐷}))
21 eqeq2 2743 . . . . . . . 8 (𝑧 = 𝐶 → (𝐴 = 𝑧𝐴 = 𝐶))
2221anbi1d 630 . . . . . . 7 (𝑧 = 𝐶 → ((𝐴 = 𝑧𝐵 = 𝐷) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
23 eqeq2 2743 . . . . . . . 8 (𝑧 = 𝐶 → (𝐵 = 𝑧𝐵 = 𝐶))
2423anbi2d 629 . . . . . . 7 (𝑧 = 𝐶 → ((𝐴 = 𝐷𝐵 = 𝑧) ↔ (𝐴 = 𝐷𝐵 = 𝐶)))
2522, 24orbi12d 917 . . . . . 6 (𝑧 = 𝐶 → (((𝐴 = 𝑧𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝑧)) ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
2620, 25bibi12d 345 . . . . 5 (𝑧 = 𝐶 → (({𝐴, 𝐵} = {𝑧, 𝐷} ↔ ((𝐴 = 𝑧𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝑧))) ↔ ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)))))
2726imbi2d 340 . . . 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 4731 . . . . . . 7 (𝑤 = 𝐷 → {𝑧, 𝑤} = {𝑧, 𝐷})
2928eqeq2d 2742 . . . . . 6 (𝑤 = 𝐷 → ({𝑥, 𝑦} = {𝑧, 𝑤} ↔ {𝑥, 𝑦} = {𝑧, 𝐷}))
30 eqeq2 2743 . . . . . . . 8 (𝑤 = 𝐷 → (𝑦 = 𝑤𝑦 = 𝐷))
3130anbi2d 629 . . . . . . 7 (𝑤 = 𝐷 → ((𝑥 = 𝑧𝑦 = 𝑤) ↔ (𝑥 = 𝑧𝑦 = 𝐷)))
32 eqeq2 2743 . . . . . . . 8 (𝑤 = 𝐷 → (𝑥 = 𝑤𝑥 = 𝐷))
3332anbi1d 630 . . . . . . 7 (𝑤 = 𝐷 → ((𝑥 = 𝑤𝑦 = 𝑧) ↔ (𝑥 = 𝐷𝑦 = 𝑧)))
3431, 33orbi12d 917 . . . . . 6 (𝑤 = 𝐷 → (((𝑥 = 𝑧𝑦 = 𝑤) ∨ (𝑥 = 𝑤𝑦 = 𝑧)) ↔ ((𝑥 = 𝑧𝑦 = 𝐷) ∨ (𝑥 = 𝐷𝑦 = 𝑧))))
35 vex 3477 . . . . . . 7 𝑥 ∈ V
36 vex 3477 . . . . . . 7 𝑦 ∈ V
37 vex 3477 . . . . . . 7 𝑧 ∈ V
38 vex 3477 . . . . . . 7 𝑤 ∈ V
3935, 36, 37, 38preq12b 4844 . . . . . 6 ({𝑥, 𝑦} = {𝑧, 𝑤} ↔ ((𝑥 = 𝑧𝑦 = 𝑤) ∨ (𝑥 = 𝑤𝑦 = 𝑧)))
4029, 34, 39vtoclbg 3556 . . . . 5 (𝐷𝑌 → ({𝑥, 𝑦} = {𝑧, 𝐷} ↔ ((𝑥 = 𝑧𝑦 = 𝐷) ∨ (𝑥 = 𝐷𝑦 = 𝑧))))
4140a1i 11 . . . 4 ((𝑥𝑉𝑦𝑊𝑧𝑋) → (𝐷𝑌 → ({𝑥, 𝑦} = {𝑧, 𝐷} ↔ ((𝑥 = 𝑧𝑦 = 𝐷) ∨ (𝑥 = 𝐷𝑦 = 𝑧)))))
429, 18, 27, 41vtocl3ga 3566 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐷𝑌 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)))))
43423expa 1118 . 2 (((𝐴𝑉𝐵𝑊) ∧ 𝐶𝑋) → (𝐷𝑌 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)))))
4443impr 455 1 (((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑋𝐷𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  {cpr 4624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-un 3949  df-sn 4623  df-pr 4625
This theorem is referenced by:  prneimg  4848  pr1eqbg  4850  preqsnd  4852  preq12nebg  4856  opthprneg  4858  preleqALT  9594  pythagtriplem2  16732  pythagtrip  16749  upgrpredgv  28264  uhgr2edg  28330  usgredg2v  28349  2pthon3v  29062  opprb  45513  or2expropbi  45516  ich2exprop  45911  prsprel  45927  paireqne  45951  poprelb  45964
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