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Theorem undif4 3284
 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 pm2.621 666 . . . . . . 7
2 olc 632 . . . . . . 7
31, 2impbid1 130 . . . . . 6
43anbi2d 437 . . . . 5
5 eldif 2927 . . . . . . 7
65orbi2i 679 . . . . . 6
7 ordi 729 . . . . . 6
86, 7bitri 173 . . . . 5
9 elun 3084 . . . . . 6
109anbi1i 431 . . . . 5
114, 8, 103bitr4g 212 . . . 4
12 elun 3084 . . . 4
13 eldif 2927 . . . 4
1411, 12, 133bitr4g 212 . . 3
1514alimi 1344 . 2
16 disj1 3270 . 2
17 dfcleq 2034 . 2
1815, 16, 173imtr4i 190 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 97   wb 98   wo 629  wal 1241   wceq 1243   wcel 1393   cdif 2914   cun 2915   cin 2916  c0 3224 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-nul 3225 This theorem is referenced by:  phplem1  6315
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