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Theorem iunxiun 4048
 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 3973 . . . . . . . 8
21anbi1i 676 . . . . . . 7
3 r19.41v 2764 . . . . . . 7
42, 3bitr4i 243 . . . . . 6
54exbii 1582 . . . . 5
6 rexcom4 2878 . . . . 5
75, 6bitr4i 243 . . . 4
8 df-rex 2620 . . . 4
9 eliun 3973 . . . . . 6
10 df-rex 2620 . . . . . 6
119, 10bitri 240 . . . . 5
1211rexbii 2639 . . . 4
137, 8, 123bitr4i 268 . . 3
14 eliun 3973 . . 3
15 eliun 3973 . . 3
1613, 14, 153bitr4i 268 . 2
1716eqriv 2350 1
 Colors of variables: wff setvar class Syntax hints:   wa 358  wex 1541   wceq 1642   wcel 1710  wrex 2615  ciun 3969 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-v 2861  df-iun 3971 This theorem is referenced by: (None)
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