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Theorem kmlem1 7792
 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 2804 . . . . . 6
21rabex 4181 . . . . 5
3 raleq 2749 . . . . . . 7
4 raleq 2749 . . . . . . . 8
54raleqbi1dv 2757 . . . . . . 7
63, 5anbi12d 691 . . . . . 6
7 raleq 2749 . . . . . . 7
87exbidv 1616 . . . . . 6
96, 8imbi12d 311 . . . . 5
102, 9spcv 2887 . . . 4
1110alrimiv 1621 . . 3
12 ssrab2 3271 . . . . . . . . 9
1312sseli 3189 . . . . . . . 8
1412sseli 3189 . . . . . . . . . 10
1514imim1i 54 . . . . . . . . 9
1615ralimi2 2628 . . . . . . . 8
1713, 16imim12i 53 . . . . . . 7
1817ralimi2 2628 . . . . . 6
19 neeq1 2467 . . . . . . . . 9
2019elrab 2936 . . . . . . . 8
2120simprbi 450 . . . . . . 7
2221rgen 2621 . . . . . 6
2318, 22jctil 523 . . . . 5
2420biimpri 197 . . . . . . . . 9
2524imim1i 54 . . . . . . . 8
2625exp3a 425 . . . . . . 7
2726ralimi2 2628 . . . . . 6
2827eximi 1566 . . . . 5
2923, 28imim12i 53 . . . 4
3029alimi 1549 . . 3
3111, 30syl 15 . 2
32 raleq 2749 . . . . 5
3332raleqbi1dv 2757 . . . 4
34 raleq 2749 . . . . 5
3534exbidv 1616 . . . 4
3633, 35imbi12d 311 . . 3
3736cbvalv 1955 . 2
3831, 37sylib 188 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358  wal 1530  wex 1531   wceq 1632   wcel 1696   wne 2459  wral 2556  crab 2560  c0 3468 This theorem is referenced by:  kmlem13  7804 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  ax-sep 4157 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-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rab 2565  df-v 2803  df-in 3172  df-ss 3179
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