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Theorem kmlem9 8586
 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 3090 . . . 4
2 eqeq1 2433 . . . . 5
32rexbidv 2946 . . . 4
4 kmlem9.1 . . . 4
51, 3, 4elab2 3227 . . 3
6 vex 3090 . . . . 5
7 eqeq1 2433 . . . . . 6
87rexbidv 2946 . . . . 5
96, 8, 4elab2 3227 . . . 4
10 difeq1 3582 . . . . . . 7
11 sneq 4012 . . . . . . . . . 10
1211difeq2d 3589 . . . . . . . . 9
1312unieqd 4232 . . . . . . . 8
1413difeq2d 3589 . . . . . . 7
1510, 14eqtrd 2470 . . . . . 6
1615eqeq2d 2443 . . . . 5
1716cbvrexv 3063 . . . 4
189, 17bitri 252 . . 3
19 reeanv 3003 . . . 4
20 eqeq12 2448 . . . . . . . . . 10
2115, 20syl5ibr 224 . . . . . . . . 9
2221necon3d 2655 . . . . . . . 8
23 kmlem5 8582 . . . . . . . . . 10
24 ineq12 3665 . . . . . . . . . . 11
2524eqeq1d 2431 . . . . . . . . . 10
2623, 25syl5ibr 224 . . . . . . . . 9
2726expd 437 . . . . . . . 8
2822, 27syl5d 69 . . . . . . 7
2928com12 32 . . . . . 6
3029adantl 467 . . . . 5
3130rexlimivv 2929 . . . 4
3219, 31sylbir 216 . . 3
335, 18, 32syl2anb 481 . 2
3433rgen2a 2859 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 370   wceq 1437   wcel 1870  cab 2414   wne 2625  wral 2782  wrex 2783   cdif 3439   cin 3441  c0 3767  csn 4002  cuni 4222 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-in 3449  df-ss 3456  df-nul 3768  df-sn 4003  df-uni 4223 This theorem is referenced by:  kmlem10  8587
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