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Theorem kmlem4 8025
 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 3322 . . . . 5
2 simpr 448 . . . . . 6
3 eluni 4010 . . . . . . . 8
43notbii 288 . . . . . . 7
5 alnex 1552 . . . . . . 7
6 con2b 325 . . . . . . . . 9
7 imnan 412 . . . . . . . . 9
8 eldifsn 3919 . . . . . . . . . . 11
9 necom 2679 . . . . . . . . . . . 12
109anbi2i 676 . . . . . . . . . . 11
118, 10bitri 241 . . . . . . . . . 10
1211imbi1i 316 . . . . . . . . 9
136, 7, 123bitr3i 267 . . . . . . . 8
1413albii 1575 . . . . . . 7
154, 5, 143bitr2i 265 . . . . . 6
162, 15sylib 189 . . . . 5
171, 16sylbi 188 . . . 4
18 eleq1 2495 . . . . . . . 8
19 neeq2 2607 . . . . . . . 8
2018, 19anbi12d 692 . . . . . . 7
21 eleq2 2496 . . . . . . . 8
2221notbid 286 . . . . . . 7
2320, 22imbi12d 312 . . . . . 6
2423spv 1965 . . . . 5
2524com12 29 . . . 4
2617, 25syl5 30 . . 3
2726ralrimiv 2780 . 2
28 disj 3660 . 2
2927, 28sylibr 204 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 359  wal 1549  wex 1550   wceq 1652   wcel 1725   wne 2598  wral 2697   cdif 3309   cin 3311  c0 3620  csn 3806  cuni 4007 This theorem is referenced by:  kmlem5  8026  kmlem11  8032 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-ne 2600  df-ral 2702  df-v 2950  df-dif 3315  df-in 3319  df-nul 3621  df-sn 3812  df-uni 4008
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