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Theorem iinin2 4400
 Description: Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4384 to recover Enderton's theorem. (Contributed by Mario Carneiro, 19-Mar-2015.)
Assertion
Ref Expression
iinin2
Distinct variable groups:   ,   ,

Proof of Theorem iinin2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 r19.28zv 3924 . . . 4
2 elin 3686 . . . . 5
32ralbii 2888 . . . 4
4 vex 3112 . . . . . 6
5 eliin 4336 . . . . . 6
64, 5ax-mp 5 . . . . 5
76anbi2i 694 . . . 4
81, 3, 73bitr4g 288 . . 3
9 eliin 4336 . . . 4
104, 9ax-mp 5 . . 3
11 elin 3686 . . 3
128, 10, 113bitr4g 288 . 2
1312eqrdv 2454 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818  =/=wne 2652  A.wral 2807   cvv 3109  i^icin 3474   c0 3784  |^|_ciin 4331 This theorem is referenced by:  iinin1  4401 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-v 3111  df-dif 3478  df-in 3482  df-nul 3785  df-iin 4333
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