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Theorem iinin2m 3725
 Description: Indexed intersection of intersection. Compare to Theorem "Distributive laws" in [Enderton] p. 30. (Contributed by Jim Kingdon, 17-Aug-2018.)
Assertion
Ref Expression
iinin2m
Distinct variable groups:   ,   ,
Allowed substitution hint:   ()

Proof of Theorem iinin2m
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 r19.28mv 3314 . . . 4
2 elin 3126 . . . . 5
32ralbii 2330 . . . 4
4 vex 2560 . . . . . 6
5 eliin 3662 . . . . . 6
64, 5ax-mp 7 . . . . 5
76anbi2i 430 . . . 4
81, 3, 73bitr4g 212 . . 3
9 eliin 3662 . . . 4
104, 9ax-mp 7 . . 3
11 elin 3126 . . 3
128, 10, 113bitr4g 212 . 2
1312eqrdv 2038 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97   wb 98   wceq 1243  wex 1381   wcel 1393  wral 2306  cvv 2557   cin 2916  ciin 3658 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-v 2559  df-in 2924  df-iin 3660 This theorem is referenced by:  iinin1m  3726
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