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Theorem kmlem4 8583
 Description: Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 26-Mar-2004.)
Assertion
Ref Expression
kmlem4
Distinct variable group:   ,,

Proof of Theorem kmlem4
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldif 3414 . . . . 5
2 simpr 463 . . . . . 6
3 eluni 4201 . . . . . . . 8
43notbii 298 . . . . . . 7
5 alnex 1665 . . . . . . 7
6 con2b 336 . . . . . . . . 9
7 imnan 424 . . . . . . . . 9
8 eldifsn 4097 . . . . . . . . . . 11
9 necom 2677 . . . . . . . . . . . 12
109anbi2i 700 . . . . . . . . . . 11
118, 10bitri 253 . . . . . . . . . 10
1211imbi1i 327 . . . . . . . . 9
136, 7, 123bitr3i 279 . . . . . . . 8
1413albii 1691 . . . . . . 7
154, 5, 143bitr2i 277 . . . . . 6
162, 15sylib 200 . . . . 5
171, 16sylbi 199 . . . 4
18 eleq1 2517 . . . . . . . 8
19 neeq2 2687 . . . . . . . 8
2018, 19anbi12d 717 . . . . . . 7
21 eleq2 2518 . . . . . . . 8
2221notbid 296 . . . . . . 7
2320, 22imbi12d 322 . . . . . 6
2423spv 2104 . . . . 5
2524com12 32 . . . 4
2617, 25syl5 33 . . 3
2726ralrimiv 2800 . 2
28 disj 3805 . 2
2927, 28sylibr 216 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wa 371  wal 1442   wceq 1444  wex 1663   wcel 1887   wne 2622  wral 2737   cdif 3401   cin 3403  c0 3731  csn 3968  cuni 4198 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431 This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-v 3047  df-dif 3407  df-in 3411  df-nul 3732  df-sn 3969  df-uni 4199 This theorem is referenced by:  kmlem5  8584  kmlem11  8590
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