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Theorem kmlem9 8559
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:   , , , ,   , ,

Proof of Theorem kmlem9
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 vex 3112 . . . 4
2 eqeq1 2461 . . . . 5
32rexbidv 2968 . . . 4
4 kmlem9.1 . . . 4
51, 3, 4elab2 3249 . . 3
6 vex 3112 . . . . 5
7 eqeq1 2461 . . . . . 6
87rexbidv 2968 . . . . 5
96, 8, 4elab2 3249 . . . 4
10 difeq1 3614 . . . . . . 7
11 sneq 4039 . . . . . . . . . 10
1211difeq2d 3621 . . . . . . . . 9
1312unieqd 4259 . . . . . . . 8
1413difeq2d 3621 . . . . . . 7
1510, 14eqtrd 2498 . . . . . 6
1615eqeq2d 2471 . . . . 5
1716cbvrexv 3085 . . . 4
189, 17bitri 249 . . 3
19 reeanv 3025 . . . 4
20 eqeq12 2476 . . . . . . . . . 10
2115, 20syl5ibr 221 . . . . . . . . 9
2221necon3d 2681 . . . . . . . 8
23 kmlem5 8555 . . . . . . . . . 10
24 ineq12 3694 . . . . . . . . . . 11
2524eqeq1d 2459 . . . . . . . . . 10
2623, 25syl5ibr 221 . . . . . . . . 9
2726expd 436 . . . . . . . 8
2822, 27syl5d 67 . . . . . . 7
2928com12 31 . . . . . 6
3029adantl 466 . . . . 5
3130rexlimivv 2954 . . . 4
3219, 31sylbir 213 . . 3
335, 18, 32syl2anb 479 . 2
3433rgen2a 2884 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  /\wa 369  =wceq 1395  e.wcel 1818  {cab 2442  =/=wne 2652  A.wral 2807  E.wrex 2808  \cdif 3472  i^icin 3474   c0 3784  {csn 4029  U.cuni 4249
This theorem is referenced by:  kmlem10  8560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-in 3482  df-ss 3489  df-nul 3785  df-sn 4030  df-uni 4250
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