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Theorem undif4 3883
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 pm2.621 408 . . . . . . 7
2 olc 384 . . . . . . 7
31, 2impbid1 203 . . . . . 6
43anbi2d 703 . . . . 5
5 eldif 3485 . . . . . . 7
65orbi2i 519 . . . . . 6
7 ordi 864 . . . . . 6
86, 7bitri 249 . . . . 5
9 elun 3644 . . . . . 6
109anbi1i 695 . . . . 5
114, 8, 103bitr4g 288 . . . 4
12 elun 3644 . . . 4
13 eldif 3485 . . . 4
1411, 12, 133bitr4g 288 . . 3
1514alimi 1633 . 2
16 disj1 3869 . 2
17 dfcleq 2450 . 2
1815, 16, 173imtr4i 266 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  <->wb 184  \/wo 368  /\wa 369  A.wal 1393  =wceq 1395  e.wcel 1818  \cdif 3472  u.cun 3473  i^icin 3474   c0 3784
This theorem is referenced by:  phplem1  7716  infdifsn  8094  difico  27594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-nul 3785
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