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Theorem prel12g 4210
Description: Closed form of prel12 4207. (Contributed by AV, 9-Dec-2018.)
Assertion
Ref Expression
prel12g

Proof of Theorem prel12g
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2461 . . . . . . 7
21notbid 294 . . . . . 6
3 preq1 4109 . . . . . . . 8
43eqeq1d 2459 . . . . . . 7
5 eleq1 2529 . . . . . . . 8
65anbi1d 704 . . . . . . 7
74, 6bibi12d 321 . . . . . 6
82, 7imbi12d 320 . . . . 5
98imbi2d 316 . . . 4
10 eqeq2 2472 . . . . . . 7
1110notbid 294 . . . . . 6
12 preq2 4110 . . . . . . . 8
1312eqeq1d 2459 . . . . . . 7
14 eleq1 2529 . . . . . . . 8
1514anbi2d 703 . . . . . . 7
1613, 15bibi12d 321 . . . . . 6
1711, 16imbi12d 320 . . . . 5
1817imbi2d 316 . . . 4
19 preq1 4109 . . . . . . . 8
2019eqeq2d 2471 . . . . . . 7
2119eleq2d 2527 . . . . . . . 8
2219eleq2d 2527 . . . . . . . 8
2321, 22anbi12d 710 . . . . . . 7
2420, 23bibi12d 321 . . . . . 6
2524imbi2d 316 . . . . 5
2625imbi2d 316 . . . 4
27 preq2 4110 . . . . . . . . 9
2827eqeq2d 2471 . . . . . . . 8
2927eleq2d 2527 . . . . . . . . 9
3027eleq2d 2527 . . . . . . . . 9
3129, 30anbi12d 710 . . . . . . . 8
3228, 31bibi12d 321 . . . . . . 7
3332imbi2d 316 . . . . . 6
34 vex 3112 . . . . . . 7
35 vex 3112 . . . . . . 7
36 vex 3112 . . . . . . 7
37 vex 3112 . . . . . . 7
3834, 35, 36, 37prel12 4207 . . . . . 6
3933, 38vtoclg 3167 . . . . 5
4039a1i 11 . . . 4
419, 18, 26, 40vtocl3ga 3177 . . 3
42413expa 1196 . 2
4342impr 619 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  <->wb 184  /\wa 369  /\w3a 973  =wceq 1395  e.wcel 1818  {cpr 4031
This theorem is referenced by:  hash2prd  12518
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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