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Theorem iinin2m 3716
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  |^|_  i^i  C  i^i  |^|_  C
Distinct variable groups:   ,   ,
Allowed substitution hint:    C()

Proof of Theorem iinin2m
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 r19.28mv 3308 . . . 4  C  C
2 elin 3120 . . . . 5  i^i  C  C
32ralbii 2324 . . . 4  i^i  C  C
4 vex 2554 . . . . . 6 
_V
5 eliin 3653 . . . . . 6  _V  |^|_  C  C
64, 5ax-mp 7 . . . . 5  |^|_  C  C
76anbi2i 430 . . . 4  |^|_  C  C
81, 3, 73bitr4g 212 . . 3  i^i  C  |^|_  C
9 eliin 3653 . . . 4  _V  |^|_  i^i  C  i^i  C
104, 9ax-mp 7 . . 3  |^|_  i^i  C  i^i  C
11 elin 3120 . . 3  i^i  |^|_  C  |^|_  C
128, 10, 113bitr4g 212 . 2  |^|_  i^i  C  i^i  |^|_  C
1312eqrdv 2035 1  |^|_  i^i  C  i^i  |^|_  C
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242  wex 1378   wcel 1390  wral 2300   _Vcvv 2551    i^i cin 2910   |^|_ciin 3649
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-bnd 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-v 2553  df-in 2918  df-iin 3651
This theorem is referenced by:  iinin1m  3717
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