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Theorem indifdir 3729
 Description: Distribute intersection over difference. (Contributed by Scott Fenton, 14-Apr-2011.)
Assertion
Ref Expression
indifdir

Proof of Theorem indifdir
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 pm3.24 890 . . . . . . . 8
21intnan 922 . . . . . . 7
3 anass 653 . . . . . . 7
42, 3mtbir 300 . . . . . 6
54biorfi 408 . . . . 5
6 an32 805 . . . . 5
7 andi 875 . . . . 5
85, 6, 73bitr4i 280 . . . 4
9 ianor 490 . . . . 5
109anbi2i 698 . . . 4
118, 10bitr4i 255 . . 3
12 elin 3649 . . . 4
13 eldif 3446 . . . . 5
1413anbi1i 699 . . . 4
1512, 14bitri 252 . . 3
16 eldif 3446 . . . 4
17 elin 3649 . . . . 5
18 elin 3649 . . . . . 6
1918notbii 297 . . . . 5
2017, 19anbi12i 701 . . . 4
2116, 20bitri 252 . . 3
2211, 15, 213bitr4i 280 . 2
2322eqriv 2418 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wo 369   wa 370   wceq 1437   wcel 1872   cdif 3433   cin 3435 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-v 3082  df-dif 3439  df-in 3443 This theorem is referenced by:  preddif  5424  fresaun  5771  uniioombllem4  22542
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