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Theorem uniuni 4183
 Description: Expression for double union that moves union into a class builder. (Contributed by FL, 28-May-2007.)
Assertion
Ref Expression
uniuni
Distinct variable group:   ,,

Proof of Theorem uniuni
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eluni 3583 . . . . . 6
21anbi2i 430 . . . . 5
32exbii 1496 . . . 4
4 19.42v 1786 . . . . . . 7
54bicomi 123 . . . . . 6
65exbii 1496 . . . . 5
7 excom 1554 . . . . . 6
8 anass 381 . . . . . . . 8
9 ancom 253 . . . . . . . 8
108, 9bitr3i 175 . . . . . . 7
11102exbii 1497 . . . . . 6
12 exdistr 1787 . . . . . 6
137, 11, 123bitri 195 . . . . 5
14 eluni 3583 . . . . . . . 8
1514bicomi 123 . . . . . . 7
1615anbi2i 430 . . . . . 6
1716exbii 1496 . . . . 5
186, 13, 173bitri 195 . . . 4
19 vex 2560 . . . . . . . . . . 11
2019uniex 4174 . . . . . . . . . 10
21 eleq2 2101 . . . . . . . . . 10
2220, 21ceqsexv 2593 . . . . . . . . 9
23 exancom 1499 . . . . . . . . 9
2422, 23bitr3i 175 . . . . . . . 8
2524anbi2i 430 . . . . . . 7
26 19.42v 1786 . . . . . . 7
27 ancom 253 . . . . . . . . 9
28 anass 381 . . . . . . . . 9
2927, 28bitri 173 . . . . . . . 8
3029exbii 1496 . . . . . . 7
3125, 26, 303bitr2i 197 . . . . . 6
3231exbii 1496 . . . . 5
33 excom 1554 . . . . 5
34 exdistr 1787 . . . . . 6
35 vex 2560 . . . . . . . . . 10
36 eqeq1 2046 . . . . . . . . . . . 12
3736anbi1d 438 . . . . . . . . . . 11
3837exbidv 1706 . . . . . . . . . 10
3935, 38elab 2687 . . . . . . . . 9
4039bicomi 123 . . . . . . . 8
4140anbi2i 430 . . . . . . 7
4241exbii 1496 . . . . . 6
4334, 42bitri 173 . . . . 5
4432, 33, 433bitri 195 . . . 4
453, 18, 443bitri 195 . . 3
4645abbii 2153 . 2
47 df-uni 3581 . 2
48 df-uni 3581 . 2
4946, 47, 483eqtr4i 2070 1
 Colors of variables: wff set class Syntax hints:   wa 97   wceq 1243  wex 1381   wcel 1393  cab 2026  cuni 3580 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-un 4170 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-uni 3581 This theorem is referenced by: (None)
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