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Theorem indifdir 3599
 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 pm3.24 854 . . . . . . . 8
21intnan 882 . . . . . . 7
3 anass 632 . . . . . . 7
42, 3mtbir 292 . . . . . 6
54biorfi 398 . . . . 5
6 an32 775 . . . . 5
7 andi 839 . . . . 5
85, 6, 73bitr4i 270 . . . 4
9 ianor 476 . . . . 5
109anbi2i 677 . . . 4
118, 10bitr4i 245 . . 3
12 elin 3532 . . . 4
13 eldif 3332 . . . . 5
1413anbi1i 678 . . . 4
1512, 14bitri 242 . . 3
16 eldif 3332 . . . 4
17 elin 3532 . . . . 5
18 elin 3532 . . . . . 6
1918notbii 289 . . . . 5
2017, 19anbi12i 680 . . . 4
2116, 20bitri 242 . . 3
2211, 15, 213bitr4i 270 . 2
2322eqriv 2435 1
 Colors of variables: wff set class Syntax hints:   wn 3   wo 359   wa 360   wceq 1653   wcel 1726   cdif 3319   cin 3321 This theorem is referenced by:  fresaun  5616  uniioombllem4  19480  preddif  25468 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-dif 3325  df-in 3329
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