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Theorem undif4 3676
 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 pm2.621 398 . . . . . . 7
2 olc 374 . . . . . . 7
31, 2impbid1 195 . . . . . 6
43anbi2d 685 . . . . 5
5 eldif 3322 . . . . . . 7
65orbi2i 506 . . . . . 6
7 ordi 835 . . . . . 6
86, 7bitri 241 . . . . 5
9 elun 3480 . . . . . 6
109anbi1i 677 . . . . 5
114, 8, 103bitr4g 280 . . . 4
12 elun 3480 . . . 4
13 eldif 3322 . . . 4
1411, 12, 133bitr4g 280 . . 3
1514alimi 1568 . 2
16 disj1 3662 . 2
17 dfcleq 2429 . 2
1815, 16, 173imtr4i 258 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 177   wo 358   wa 359  wal 1549   wceq 1652   wcel 1725   cdif 3309   cun 3310   cin 3311  c0 3620 This theorem is referenced by:  phplem1  7278  infdifsn  7603  difico  24138 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-nul 3621
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