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Theorem kmlem4 8554
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 3485 . . . . 5
2 simpr 461 . . . . . 6
3 eluni 4252 . . . . . . . 8
43notbii 296 . . . . . . 7
5 alnex 1614 . . . . . . 7
6 con2b 334 . . . . . . . . 9
7 imnan 422 . . . . . . . . 9
8 eldifsn 4155 . . . . . . . . . . 11
9 necom 2726 . . . . . . . . . . . 12
109anbi2i 694 . . . . . . . . . . 11
118, 10bitri 249 . . . . . . . . . 10
1211imbi1i 325 . . . . . . . . 9
136, 7, 123bitr3i 275 . . . . . . . 8
1413albii 1640 . . . . . . 7
154, 5, 143bitr2i 273 . . . . . 6
162, 15sylib 196 . . . . 5
171, 16sylbi 195 . . . 4
18 eleq1 2529 . . . . . . . 8
19 neeq2 2740 . . . . . . . 8
2018, 19anbi12d 710 . . . . . . 7
21 eleq2 2530 . . . . . . . 8
2221notbid 294 . . . . . . 7
2320, 22imbi12d 320 . . . . . 6
2423spv 2011 . . . . 5
2524com12 31 . . . 4
2617, 25syl5 32 . . 3
2726ralrimiv 2869 . 2
28 disj 3867 . 2
2927, 28sylibr 212 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  /\wa 369  A.wal 1393  =wceq 1395  E.wex 1612  e.wcel 1818  =/=wne 2652  A.wral 2807  \cdif 3472  i^icin 3474   c0 3784  {csn 4029  U.cuni 4249
This theorem is referenced by:  kmlem5  8555  kmlem11  8561
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-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-ne 2654  df-ral 2812  df-v 3111  df-dif 3478  df-in 3482  df-nul 3785  df-sn 4030  df-uni 4250
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