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Theorem iunxiun 3727
Description: Separate an indexed union in the index of an indexed union. (Contributed by Mario Carneiro, 5-Dec-2016.)
Assertion
Ref Expression
iunxiun  U_  U_  C  U_  U_  C
Distinct variable groups:   ,   ,   , C
Allowed substitution hints:   ()   (,)    C()

Proof of Theorem iunxiun
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eliun 3652 . . . . . . . 8  U_
21anbi1i 431 . . . . . . 7  U_  C  C
3 r19.41v 2460 . . . . . . 7  C  C
42, 3bitr4i 176 . . . . . 6  U_  C  C
54exbii 1493 . . . . 5  U_  C  C
6 rexcom4 2571 . . . . 5  C  C
75, 6bitr4i 176 . . . 4  U_  C  C
8 df-rex 2306 . . . 4  U_  C  U_  C
9 eliun 3652 . . . . . 6  U_  C  C
10 df-rex 2306 . . . . . 6  C  C
119, 10bitri 173 . . . . 5  U_  C  C
1211rexbii 2325 . . . 4  U_  C  C
137, 8, 123bitr4i 201 . . 3  U_  C  U_  C
14 eliun 3652 . . 3  U_ 
U_  C  U_  C
15 eliun 3652 . . 3  U_  U_  C  U_  C
1613, 14, 153bitr4i 201 . 2  U_ 
U_  C  U_  U_  C
1716eqriv 2034 1  U_  U_  C  U_  U_  C
Colors of variables: wff set class
Syntax hints:   wa 97   wceq 1242  wex 1378   wcel 1390  wrex 2301   U_ciun 3648
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-iun 3650
This theorem is referenced by: (None)
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