Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  preq12bg Structured version   Visualization version   GIF version

Theorem preq12bg 4530
 Description: Closed form of preq12b 4526. (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 4412 . . . . . . 7 (𝑥 = 𝐴 → {𝑥, 𝑦} = {𝐴, 𝑦})
21eqeq1d 2762 . . . . . 6 (𝑥 = 𝐴 → ({𝑥, 𝑦} = {𝑧, 𝐷} ↔ {𝐴, 𝑦} = {𝑧, 𝐷}))
3 eqeq1 2764 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥 = 𝑧𝐴 = 𝑧))
43anbi1d 743 . . . . . . 7 (𝑥 = 𝐴 → ((𝑥 = 𝑧𝑦 = 𝐷) ↔ (𝐴 = 𝑧𝑦 = 𝐷)))
5 eqeq1 2764 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥 = 𝐷𝐴 = 𝐷))
65anbi1d 743 . . . . . . 7 (𝑥 = 𝐴 → ((𝑥 = 𝐷𝑦 = 𝑧) ↔ (𝐴 = 𝐷𝑦 = 𝑧)))
74, 6orbi12d 748 . . . . . 6 (𝑥 = 𝐴 → (((𝑥 = 𝑧𝑦 = 𝐷) ∨ (𝑥 = 𝐷𝑦 = 𝑧)) ↔ ((𝐴 = 𝑧𝑦 = 𝐷) ∨ (𝐴 = 𝐷𝑦 = 𝑧))))
82, 7bibi12d 334 . . . . 5 (𝑥 = 𝐴 → (({𝑥, 𝑦} = {𝑧, 𝐷} ↔ ((𝑥 = 𝑧𝑦 = 𝐷) ∨ (𝑥 = 𝐷𝑦 = 𝑧))) ↔ ({𝐴, 𝑦} = {𝑧, 𝐷} ↔ ((𝐴 = 𝑧𝑦 = 𝐷) ∨ (𝐴 = 𝐷𝑦 = 𝑧)))))
98imbi2d 329 . . . 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 4413 . . . . . . 7 (𝑦 = 𝐵 → {𝐴, 𝑦} = {𝐴, 𝐵})
1110eqeq1d 2762 . . . . . 6 (𝑦 = 𝐵 → ({𝐴, 𝑦} = {𝑧, 𝐷} ↔ {𝐴, 𝐵} = {𝑧, 𝐷}))
12 eqeq1 2764 . . . . . . . 8 (𝑦 = 𝐵 → (𝑦 = 𝐷𝐵 = 𝐷))
1312anbi2d 742 . . . . . . 7 (𝑦 = 𝐵 → ((𝐴 = 𝑧𝑦 = 𝐷) ↔ (𝐴 = 𝑧𝐵 = 𝐷)))
14 eqeq1 2764 . . . . . . . 8 (𝑦 = 𝐵 → (𝑦 = 𝑧𝐵 = 𝑧))
1514anbi2d 742 . . . . . . 7 (𝑦 = 𝐵 → ((𝐴 = 𝐷𝑦 = 𝑧) ↔ (𝐴 = 𝐷𝐵 = 𝑧)))
1613, 15orbi12d 748 . . . . . 6 (𝑦 = 𝐵 → (((𝐴 = 𝑧𝑦 = 𝐷) ∨ (𝐴 = 𝐷𝑦 = 𝑧)) ↔ ((𝐴 = 𝑧𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝑧))))
1711, 16bibi12d 334 . . . . 5 (𝑦 = 𝐵 → (({𝐴, 𝑦} = {𝑧, 𝐷} ↔ ((𝐴 = 𝑧𝑦 = 𝐷) ∨ (𝐴 = 𝐷𝑦 = 𝑧))) ↔ ({𝐴, 𝐵} = {𝑧, 𝐷} ↔ ((𝐴 = 𝑧𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝑧)))))
1817imbi2d 329 . . . 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 4412 . . . . . . 7 (𝑧 = 𝐶 → {𝑧, 𝐷} = {𝐶, 𝐷})
2019eqeq2d 2770 . . . . . 6 (𝑧 = 𝐶 → ({𝐴, 𝐵} = {𝑧, 𝐷} ↔ {𝐴, 𝐵} = {𝐶, 𝐷}))
21 eqeq2 2771 . . . . . . . 8 (𝑧 = 𝐶 → (𝐴 = 𝑧𝐴 = 𝐶))
2221anbi1d 743 . . . . . . 7 (𝑧 = 𝐶 → ((𝐴 = 𝑧𝐵 = 𝐷) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
23 eqeq2 2771 . . . . . . . 8 (𝑧 = 𝐶 → (𝐵 = 𝑧𝐵 = 𝐶))
2423anbi2d 742 . . . . . . 7 (𝑧 = 𝐶 → ((𝐴 = 𝐷𝐵 = 𝑧) ↔ (𝐴 = 𝐷𝐵 = 𝐶)))
2522, 24orbi12d 748 . . . . . 6 (𝑧 = 𝐶 → (((𝐴 = 𝑧𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝑧)) ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
2620, 25bibi12d 334 . . . . 5 (𝑧 = 𝐶 → (({𝐴, 𝐵} = {𝑧, 𝐷} ↔ ((𝐴 = 𝑧𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝑧))) ↔ ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)))))
2726imbi2d 329 . . . 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 4413 . . . . . . 7 (𝑤 = 𝐷 → {𝑧, 𝑤} = {𝑧, 𝐷})
2928eqeq2d 2770 . . . . . 6 (𝑤 = 𝐷 → ({𝑥, 𝑦} = {𝑧, 𝑤} ↔ {𝑥, 𝑦} = {𝑧, 𝐷}))
30 eqeq2 2771 . . . . . . . 8 (𝑤 = 𝐷 → (𝑦 = 𝑤𝑦 = 𝐷))
3130anbi2d 742 . . . . . . 7 (𝑤 = 𝐷 → ((𝑥 = 𝑧𝑦 = 𝑤) ↔ (𝑥 = 𝑧𝑦 = 𝐷)))
32 eqeq2 2771 . . . . . . . 8 (𝑤 = 𝐷 → (𝑥 = 𝑤𝑥 = 𝐷))
3332anbi1d 743 . . . . . . 7 (𝑤 = 𝐷 → ((𝑥 = 𝑤𝑦 = 𝑧) ↔ (𝑥 = 𝐷𝑦 = 𝑧)))
3431, 33orbi12d 748 . . . . . 6 (𝑤 = 𝐷 → (((𝑥 = 𝑧𝑦 = 𝑤) ∨ (𝑥 = 𝑤𝑦 = 𝑧)) ↔ ((𝑥 = 𝑧𝑦 = 𝐷) ∨ (𝑥 = 𝐷𝑦 = 𝑧))))
35 vex 3343 . . . . . . 7 𝑥 ∈ V
36 vex 3343 . . . . . . 7 𝑦 ∈ V
37 vex 3343 . . . . . . 7 𝑧 ∈ V
38 vex 3343 . . . . . . 7 𝑤 ∈ V
3935, 36, 37, 38preq12b 4526 . . . . . 6 ({𝑥, 𝑦} = {𝑧, 𝑤} ↔ ((𝑥 = 𝑧𝑦 = 𝑤) ∨ (𝑥 = 𝑤𝑦 = 𝑧)))
4029, 34, 39vtoclbg 3407 . . . . 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 3416 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐷𝑌 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)))))
43423expa 1112 . 2 (((𝐴𝑉𝐵𝑊) ∧ 𝐶𝑋) → (𝐷𝑌 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)))))
4443impr 650 1 (((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑋𝐷𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 382   ∧ wa 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139  {cpr 4323 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-un 3720  df-sn 4322  df-pr 4324 This theorem is referenced by:  prneimg  4532  pr1eqbg  4534  preqsnd  4536  preqsndOLD  4537  preq12nebg  4543  opthprneg  4545  preleqALT  8685  pythagtriplem2  15724  pythagtrip  15741  upgrpredgv  26233  uhgr2edg  26299  usgredg2v  26318  2pthon3v  27063  prsprel  42247
 Copyright terms: Public domain W3C validator