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Theorem iinin2 4036
 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
iinin2
Distinct variable groups:   ,   ,
Allowed substitution hint:   ()

Proof of Theorem iinin2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 r19.28zv 3645 . . . 4
2 elin 3219 . . . . 5
32ralbii 2638 . . . 4
4 vex 2862 . . . . . 6
5 eliin 3974 . . . . . 6
64, 5ax-mp 8 . . . . 5
76anbi2i 675 . . . 4
81, 3, 73bitr4g 279 . . 3
9 eliin 3974 . . . 4
104, 9ax-mp 8 . . 3
11 elin 3219 . . 3
128, 10, 113bitr4g 279 . 2
1312eqrdv 2351 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 176   wa 358   wceq 1642   wcel 1710   wne 2516  wral 2614  cvv 2859   cin 3208  c0 3550  ciin 3970 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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:  iinin1  4037
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