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Theorem iinin1 3871
 Description: Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 3854 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 3870 . 2
2 incom 3269 . . . 4
32a1i 12 . . 3
43iineq2i 3822 . 2
5 incom 3269 . 2
61, 4, 53eqtr4g 2310 1
 Colors of variables: wff set class Syntax hints:   wi 6   wceq 1619   wcel 1621   wne 2412   cin 3077  c0 3362  ciin 3804 This theorem is referenced by:  firest  13211 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-v 2729  df-dif 3081  df-in 3085  df-nul 3363  df-iin 3806
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