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Theorem iinin1 4154
 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
iinin1
Distinct variable groups:   ,   ,
Allowed substitution hint:   ()

Proof of Theorem iinin1
StepHypRef Expression
1 iinin2 4153 . 2
2 incom 3525 . . . 4
32a1i 11 . . 3
43iineq2i 4104 . 2
5 incom 3525 . 2
61, 4, 53eqtr4g 2492 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1652   wcel 1725   wne 2598   cin 3311  c0 3620  ciin 4086 This theorem is referenced by:  firest  13652 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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