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Theorem iinin1 4364
 Description: Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4347 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 4363 . 2
2 incom 3652 . . . 4
32a1i 11 . . 3
43iineq2i 4313 . 2
5 incom 3652 . 2
61, 4, 53eqtr4g 2486 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wceq 1437   wcel 1867   wne 2616   cin 3432  c0 3758  ciin 4294 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-v 3080  df-dif 3436  df-in 3440  df-nul 3759  df-iin 4296 This theorem is referenced by:  firest  15309
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