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Theorem iunin2 3720
 Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3710 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.)
Assertion
Ref Expression
iunin2
Distinct variable group:   ,
Allowed substitution hints:   ()   ()

Proof of Theorem iunin2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 r19.42v 2467 . . . 4
2 elin 3126 . . . . 5
32rexbii 2331 . . . 4
4 eliun 3661 . . . . 5
54anbi2i 430 . . . 4
61, 3, 53bitr4i 201 . . 3
7 eliun 3661 . . 3
8 elin 3126 . . 3
96, 7, 83bitr4i 201 . 2
109eqriv 2037 1
 Colors of variables: wff set class Syntax hints:   wa 97   wceq 1243   wcel 1393  wrex 2307   cin 2916  ciun 3657 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-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 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-ral 2311  df-rex 2312  df-v 2559  df-in 2924  df-iun 3659 This theorem is referenced by:  iunin1  3721  2iunin  3723  resiun1  4630  resiun2  4631
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