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Theorem kmlem15 7806
 Description: Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 5 <=> 4. (Contributed by NM, 4-Apr-2004.)
Hypotheses
Ref Expression
kmlem14.1
kmlem14.2
kmlem14.3
Assertion
Ref Expression
kmlem15
Distinct variable groups:   ,,,,   ,
Allowed substitution hints:   (,,,)   (,,,,)   (,,,,)

Proof of Theorem kmlem15
StepHypRef Expression
1 kmlem14.3 . . . 4
2 nfv 1609 . . . . . . 7
32eu1 2177 . . . . . 6
4 elin 3371 . . . . . . . . 9
5 clelsb3 2398 . . . . . . . . . . . 12
6 elin 3371 . . . . . . . . . . . 12
75, 6bitri 240 . . . . . . . . . . 11
8 eqcom 2298 . . . . . . . . . . 11
97, 8imbi12i 316 . . . . . . . . . 10
109albii 1556 . . . . . . . . 9
114, 10anbi12i 678 . . . . . . . 8
12 19.28v 1848 . . . . . . . 8
1311, 12bitr4i 243 . . . . . . 7
1413exbii 1572 . . . . . 6
153, 14bitri 240 . . . . 5
1615ralbii 2580 . . . 4
17 df-ral 2561 . . . . 5
18 kmlem14.2 . . . . . . . . . 10
1918albii 1556 . . . . . . . . 9
20 19.21v 1843 . . . . . . . . 9
2119, 20bitri 240 . . . . . . . 8
2221exbii 1572 . . . . . . 7
23 19.37v 1852 . . . . . . 7
2422, 23bitri 240 . . . . . 6
2524albii 1556 . . . . 5
2617, 25bitr4i 243 . . . 4
271, 16, 263bitri 262 . . 3
2827anbi2i 675 . 2
29 19.28v 1848 . 2
30 19.28v 1848 . . . . 5
3130exbii 1572 . . . 4
32 19.42v 1858 . . . 4
3331, 32bitr2i 241 . . 3
3433albii 1556 . 2
3528, 29, 343bitr2i 264 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 176   wa 358  wal 1530  wex 1531   wceq 1632  wsb 1638   wcel 1696  weu 2156   wne 2459  wral 2556   cin 3164 This theorem is referenced by:  kmlem16  7807 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-v 2803  df-in 3172
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