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Theorem indifdir 3759
 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 pm3.24 880 . . . . . . . 8
21intnan 912 . . . . . . 7
3 anass 649 . . . . . . 7
42, 3mtbir 299 . . . . . 6
54biorfi 407 . . . . 5
6 an32 796 . . . . 5
7 andi 865 . . . . 5
85, 6, 73bitr4i 277 . . . 4
9 ianor 488 . . . . 5
109anbi2i 694 . . . 4
118, 10bitr4i 252 . . 3
12 elin 3692 . . . 4
13 eldif 3491 . . . . 5
1413anbi1i 695 . . . 4
1512, 14bitri 249 . . 3
16 eldif 3491 . . . 4
17 elin 3692 . . . . 5
18 elin 3692 . . . . . 6
1918notbii 296 . . . . 5
2017, 19anbi12i 697 . . . 4
2116, 20bitri 249 . . 3
2211, 15, 213bitr4i 277 . 2
2322eqriv 2463 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wo 368   wa 369   wceq 1379   wcel 1767   cdif 3478   cin 3480 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3120  df-dif 3484  df-in 3488 This theorem is referenced by:  fresaun  5762  uniioombllem4  21863  preddif  29198
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