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Theorem iunin2 3711
 Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3701 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 2461 . . . 4
2 elin 3120 . . . . 5
32rexbii 2325 . . . 4
4 eliun 3652 . . . . 5
54anbi2i 430 . . . 4
61, 3, 53bitr4i 201 . . 3
7 eliun 3652 . . 3
8 elin 3120 . . 3
96, 7, 83bitr4i 201 . 2
109eqriv 2034 1
 Colors of variables: wff set class Syntax hints:   wa 97   wceq 1242   wcel 1390  wrex 2301   cin 2910  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-in 2918  df-iun 3650 This theorem is referenced by:  iunin1  3712  2iunin  3714  resiun1  4573  resiun2  4574
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