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Theorem iunxiun 4175
 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 4099 . . . . . . . 8
21anbi1i 678 . . . . . . 7
3 r19.41v 2863 . . . . . . 7
42, 3bitr4i 245 . . . . . 6
54exbii 1593 . . . . 5
6 rexcom4 2977 . . . . 5
75, 6bitr4i 245 . . . 4
8 df-rex 2713 . . . 4
9 eliun 4099 . . . . . 6
10 df-rex 2713 . . . . . 6
119, 10bitri 242 . . . . 5
1211rexbii 2732 . . . 4
137, 8, 123bitr4i 270 . . 3
14 eliun 4099 . . 3
15 eliun 4099 . . 3
1613, 14, 153bitr4i 270 . 2
1716eqriv 2435 1
 Colors of variables: wff set class Syntax hints:   wa 360  wex 1551   wceq 1653   wcel 1726  wrex 2708  ciun 4095 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-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-v 2960  df-iun 4097
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