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Theorem kmlem9 8030
 Description: Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 25-Mar-2004.)
Hypothesis
Ref Expression
kmlem9.1
Assertion
Ref Expression
kmlem9
Distinct variable groups:   ,,,,   ,,
Allowed substitution hints:   (,,)

Proof of Theorem kmlem9
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 vex 2951 . . . 4
2 eqeq1 2441 . . . . 5
32rexbidv 2718 . . . 4
4 kmlem9.1 . . . 4
51, 3, 4elab2 3077 . . 3
6 vex 2951 . . . . 5
7 eqeq1 2441 . . . . . 6
87rexbidv 2718 . . . . 5
96, 8, 4elab2 3077 . . . 4
10 difeq1 3450 . . . . . . 7
11 sneq 3817 . . . . . . . . . 10
1211difeq2d 3457 . . . . . . . . 9
1312unieqd 4018 . . . . . . . 8
1413difeq2d 3457 . . . . . . 7
1510, 14eqtrd 2467 . . . . . 6
1615eqeq2d 2446 . . . . 5
1716cbvrexv 2925 . . . 4
189, 17bitri 241 . . 3
19 reeanv 2867 . . . 4
20 eqeq12 2447 . . . . . . . . . 10
2115, 20syl5ibr 213 . . . . . . . . 9
2221necon3d 2636 . . . . . . . 8
23 kmlem5 8026 . . . . . . . . . 10
24 ineq12 3529 . . . . . . . . . . 11
2524eqeq1d 2443 . . . . . . . . . 10
2623, 25syl5ibr 213 . . . . . . . . 9
2726exp3a 426 . . . . . . . 8
2822, 27syl5d 64 . . . . . . 7
2928com12 29 . . . . . 6
3029adantl 453 . . . . 5
3130rexlimivv 2827 . . . 4
3219, 31sylbir 205 . . 3
335, 18, 32syl2anb 466 . 2
3433rgen2a 2764 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1652   wcel 1725  cab 2421   wne 2598  wral 2697  wrex 2698   cdif 3309   cin 3311  c0 3620  csn 3806  cuni 4007 This theorem is referenced by:  kmlem10  8031 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-in 3319  df-ss 3326  df-nul 3621  df-sn 3812  df-uni 4008
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