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Theorem iinin2 4153
 Description: Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4137 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 3715 . . . 4
2 elin 3522 . . . . 5
32ralbii 2721 . . . 4
4 vex 2951 . . . . . 6
5 eliin 4090 . . . . . 6
64, 5ax-mp 8 . . . . 5
76anbi2i 676 . . . 4
81, 3, 73bitr4g 280 . . 3
9 eliin 4090 . . . 4
104, 9ax-mp 8 . . 3
11 elin 3522 . . 3
128, 10, 113bitr4g 280 . 2
1312eqrdv 2433 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725   wne 2598  wral 2697  cvv 2948   cin 3311  c0 3620  ciin 4086 This theorem is referenced by:  iinin1  4154 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-v 2950  df-dif 3315  df-in 3319  df-nul 3621  df-iin 4088
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