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Theorem iunxiun 3765
 Description: Separate an indexed union in the index of an indexed union. (Contributed by Mario Carneiro, 5-Dec-2016.)
Assertion
Ref Expression
iunxiun
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   ()   (,)   ()

Proof of Theorem iunxiun
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eliun 3690 . . . . . . . 8
21anbi1i 446 . . . . . . 7
3 r19.41v 2511 . . . . . . 7
42, 3bitr4i 185 . . . . . 6
54exbii 1537 . . . . 5
6 rexcom4 2623 . . . . 5
75, 6bitr4i 185 . . . 4
8 df-rex 2355 . . . 4
9 eliun 3690 . . . . . 6
10 df-rex 2355 . . . . . 6
119, 10bitri 182 . . . . 5
1211rexbii 2374 . . . 4
137, 8, 123bitr4i 210 . . 3
14 eliun 3690 . . 3
15 eliun 3690 . . 3
1613, 14, 153bitr4i 210 . 2
1716eqriv 2079 1
 Colors of variables: wff set class Syntax hints:   wa 102   wceq 1285  wex 1422   wcel 1434  wrex 2350  ciun 3686 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-iun 3688 This theorem is referenced by: (None)
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