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Theorem iinin1 4037
 Description: Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4020 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 4036 . 2
2 incom 3448 . . . 4
32a1i 10 . . 3
43iineq2i 3988 . 2
5 incom 3448 . 2
61, 4, 53eqtr4g 2410 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wceq 1642   wcel 1710   wne 2516   cin 3208  c0 3550  ciin 3970 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-nul 3551  df-iin 3972 This theorem is referenced by: (None)
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