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Theorem preq12bg 3979
 Description: Closed form of preq12b 3976. (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 3885 . . . . . . 7
21eqeq1d 2446 . . . . . 6
3 eqeq1 2444 . . . . . . . 8
43anbi1d 687 . . . . . . 7
5 eqeq1 2444 . . . . . . . 8
65anbi1d 687 . . . . . . 7
74, 6orbi12d 692 . . . . . 6
82, 7bibi12d 314 . . . . 5
98imbi2d 309 . . . 4
10 preq2 3886 . . . . . . 7
1110eqeq1d 2446 . . . . . 6
12 eqeq1 2444 . . . . . . . 8
1312anbi2d 686 . . . . . . 7
14 eqeq1 2444 . . . . . . . 8
1514anbi2d 686 . . . . . . 7
1613, 15orbi12d 692 . . . . . 6
1711, 16bibi12d 314 . . . . 5
1817imbi2d 309 . . . 4
19 preq1 3885 . . . . . . 7
2019eqeq2d 2449 . . . . . 6
21 eqeq2 2447 . . . . . . . 8
2221anbi1d 687 . . . . . . 7
23 eqeq2 2447 . . . . . . . 8
2423anbi2d 686 . . . . . . 7
2522, 24orbi12d 692 . . . . . 6
2620, 25bibi12d 314 . . . . 5
2726imbi2d 309 . . . 4
28 preq2 3886 . . . . . . 7
2928eqeq2d 2449 . . . . . 6
30 eqeq2 2447 . . . . . . . 8
3130anbi2d 686 . . . . . . 7
32 eqeq2 2447 . . . . . . . 8
3332anbi1d 687 . . . . . . 7
3431, 33orbi12d 692 . . . . . 6
35 vex 2961 . . . . . . 7
36 vex 2961 . . . . . . 7
37 vex 2961 . . . . . . 7
38 vex 2961 . . . . . . 7
3935, 36, 37, 38preq12b 3976 . . . . . 6
4029, 34, 39vtoclbg 3014 . . . . 5
4140a1i 11 . . . 4
429, 18, 27, 41vtocl3ga 3023 . . 3
43423expa 1154 . 2
4443impr 604 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wo 359   wa 360   w3a 937   wceq 1653   wcel 1726  cpr 3817 This theorem is referenced by:  prneimg  3980  pythagtriplem2  13193  pythagtrip  13210  usgraidx2v  21414  constr3trllem2  21640  preqsnd  24002  pr1eqbg  28058  usgra2wlkspthlem1  28332  usgra2adedgspthlem2  28340 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-un 3327  df-sn 3822  df-pr 3823
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