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Theorem undif4 3456
 Description: Distribute union over difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
undif4

Proof of Theorem undif4
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 pm2.621 737 . . . . . . 7
2 olc 701 . . . . . . 7
31, 2impbid1 141 . . . . . 6
43anbi2d 460 . . . . 5
5 eldif 3111 . . . . . . 7
65orbi2i 752 . . . . . 6
7 ordi 806 . . . . . 6
86, 7bitri 183 . . . . 5
9 elun 3248 . . . . . 6
109anbi1i 454 . . . . 5
114, 8, 103bitr4g 222 . . . 4
12 elun 3248 . . . 4
13 eldif 3111 . . . 4
1411, 12, 133bitr4g 222 . . 3
1514alimi 1435 . 2
16 disj1 3444 . 2
17 dfcleq 2151 . 2
1815, 16, 173imtr4i 200 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 103   wb 104   wo 698  wal 1333   wceq 1335   wcel 2128   cdif 3099   cun 3100   cin 3101  c0 3394 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-nul 3395 This theorem is referenced by:  phplem1  6794
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