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Theorem iinin2 4369
 Description: Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4353 to recover Enderton's theorem. (Contributed by Mario Carneiro, 19-Mar-2015.)
Assertion
Ref Expression
iinin2
Distinct variable groups:   ,   ,
Allowed substitution hint:   ()

Proof of Theorem iinin2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 r19.28zv 3894 . . . 4
2 elin 3649 . . . . 5
32ralbii 2853 . . . 4
4 vex 3083 . . . . . 6
5 eliin 4305 . . . . . 6
64, 5ax-mp 5 . . . . 5
76anbi2i 698 . . . 4
81, 3, 73bitr4g 291 . . 3
9 eliin 4305 . . . 4
104, 9ax-mp 5 . . 3
11 elin 3649 . . 3
128, 10, 113bitr4g 291 . 2
1312eqrdv 2419 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 187   wa 370   wceq 1437   wcel 1872   wne 2614  wral 2771  cvv 3080   cin 3435  c0 3761  ciin 4300 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-v 3082  df-dif 3439  df-in 3443  df-nul 3762  df-iin 4302 This theorem is referenced by:  iinin1  4370
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