Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  kmlem1 Structured version   Unicode version

Theorem kmlem1 8035
 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 2961 . . . . . 6
21rabex 4357 . . . . 5
3 raleq 2906 . . . . . . 7
4 raleq 2906 . . . . . . . 8
54raleqbi1dv 2914 . . . . . . 7
63, 5anbi12d 693 . . . . . 6
7 raleq 2906 . . . . . . 7
87exbidv 1637 . . . . . 6
96, 8imbi12d 313 . . . . 5
102, 9spcv 3044 . . . 4
1110alrimiv 1642 . . 3
12 elrabi 3092 . . . . . . . 8
13 elrabi 3092 . . . . . . . . . 10
1413imim1i 57 . . . . . . . . 9
1514ralimi2 2780 . . . . . . . 8
1612, 15imim12i 56 . . . . . . 7
1716ralimi2 2780 . . . . . 6
18 neeq1 2611 . . . . . . . . 9
1918elrab 3094 . . . . . . . 8
2019simprbi 452 . . . . . . 7
2120rgen 2773 . . . . . 6
2217, 21jctil 525 . . . . 5
2319biimpri 199 . . . . . . . . 9
2423imim1i 57 . . . . . . . 8
2524exp3a 427 . . . . . . 7
2625ralimi2 2780 . . . . . 6
2726eximi 1586 . . . . 5
2822, 27imim12i 56 . . . 4
2928alimi 1569 . . 3
3011, 29syl 16 . 2
31 raleq 2906 . . . . 5
3231raleqbi1dv 2914 . . . 4
33 raleq 2906 . . . . 5
3433exbidv 1637 . . . 4
3532, 34imbi12d 313 . . 3
3635cbvalv 1985 . 2
3730, 36sylib 190 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360  wal 1550  wex 1551   wceq 1653   wcel 1726   wne 2601  wral 2707  crab 2711  c0 3630 This theorem is referenced by:  kmlem13  8047 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rab 2716  df-v 2960  df-in 3329  df-ss 3336
 Copyright terms: Public domain W3C validator