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Theorem kmlem1 7986
 Description: Lemma for 5-quantifier AC of Kurt Maes, Th. 4, 1 => 2. (Contributed by NM, 5-Apr-2004.)
Assertion
Ref Expression
kmlem1
Distinct variable groups:   ,,   ,   ,,,
Allowed substitution hints:   (,)   (,,)

Proof of Theorem kmlem1
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2919 . . . . . 6
21rabex 4314 . . . . 5
3 raleq 2864 . . . . . . 7
4 raleq 2864 . . . . . . . 8
54raleqbi1dv 2872 . . . . . . 7
63, 5anbi12d 692 . . . . . 6
7 raleq 2864 . . . . . . 7
87exbidv 1633 . . . . . 6
96, 8imbi12d 312 . . . . 5
102, 9spcv 3002 . . . 4
1110alrimiv 1638 . . 3
12 elrabi 3050 . . . . . . . 8
13 elrabi 3050 . . . . . . . . . 10
1413imim1i 56 . . . . . . . . 9
1514ralimi2 2738 . . . . . . . 8
1612, 15imim12i 55 . . . . . . 7
1716ralimi2 2738 . . . . . 6
18 neeq1 2575 . . . . . . . . 9
1918elrab 3052 . . . . . . . 8
2019simprbi 451 . . . . . . 7
2120rgen 2731 . . . . . 6
2217, 21jctil 524 . . . . 5
2319biimpri 198 . . . . . . . . 9
2423imim1i 56 . . . . . . . 8
2524exp3a 426 . . . . . . 7
2625ralimi2 2738 . . . . . 6
2726eximi 1582 . . . . 5
2822, 27imim12i 55 . . . 4
2928alimi 1565 . . 3
3011, 29syl 16 . 2
31 raleq 2864 . . . . 5
3231raleqbi1dv 2872 . . . 4
33 raleq 2864 . . . . 5
3433exbidv 1633 . . . 4
3532, 34imbi12d 312 . . 3
3635cbvalv 2052 . 2
3730, 36sylib 189 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359  wal 1546  wex 1547   wceq 1649   wcel 1721   wne 2567  wral 2666  crab 2670  c0 3588 This theorem is referenced by:  kmlem13  7998 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290 This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rab 2675  df-v 2918  df-in 3287  df-ss 3294
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