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Theorem preq12bg 4209
Description: Closed form of preq12b 4206. (Contributed by Scott Fenton, 28-Mar-2014.)
Assertion
Ref Expression
preq12bg

Proof of Theorem preq12bg
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 preq1 4109 . . . . . . 7
21eqeq1d 2459 . . . . . 6
3 eqeq1 2461 . . . . . . . 8
43anbi1d 704 . . . . . . 7
5 eqeq1 2461 . . . . . . . 8
65anbi1d 704 . . . . . . 7
74, 6orbi12d 709 . . . . . 6
82, 7bibi12d 321 . . . . 5
98imbi2d 316 . . . 4
10 preq2 4110 . . . . . . 7
1110eqeq1d 2459 . . . . . 6
12 eqeq1 2461 . . . . . . . 8
1312anbi2d 703 . . . . . . 7
14 eqeq1 2461 . . . . . . . 8
1514anbi2d 703 . . . . . . 7
1613, 15orbi12d 709 . . . . . 6
1711, 16bibi12d 321 . . . . 5
1817imbi2d 316 . . . 4
19 preq1 4109 . . . . . . 7
2019eqeq2d 2471 . . . . . 6
21 eqeq2 2472 . . . . . . . 8
2221anbi1d 704 . . . . . . 7
23 eqeq2 2472 . . . . . . . 8
2423anbi2d 703 . . . . . . 7
2522, 24orbi12d 709 . . . . . 6
2620, 25bibi12d 321 . . . . 5
2726imbi2d 316 . . . 4
28 preq2 4110 . . . . . . 7
2928eqeq2d 2471 . . . . . 6
30 eqeq2 2472 . . . . . . . 8
3130anbi2d 703 . . . . . . 7
32 eqeq2 2472 . . . . . . . 8
3332anbi1d 704 . . . . . . 7
3431, 33orbi12d 709 . . . . . 6
35 vex 3112 . . . . . . 7
36 vex 3112 . . . . . . 7
37 vex 3112 . . . . . . 7
38 vex 3112 . . . . . . 7
3935, 36, 37, 38preq12b 4206 . . . . . 6
4029, 34, 39vtoclbg 3168 . . . . 5
4140a1i 11 . . . 4
429, 18, 27, 41vtocl3ga 3177 . . 3
43423expa 1196 . 2
4443impr 619 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  \/wo 368  /\wa 369  /\w3a 973  =wceq 1395  e.wcel 1818  {cpr 4031
This theorem is referenced by:  prneimg  4211  pythagtriplem2  14341  pythagtrip  14358  usgraidx2v  24393  usgra2adedgspthlem2  24612  usgra2wlkspthlem1  24619  constr3trllem2  24651  preqsnd  27418  pr1eqbg  32297  usgedgvadf1  32415  usgedgvadf1ALT  32418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-un 3480  df-sn 4030  df-pr 4032
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