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Theorem hausnei 17430
 Description: Neighborhood property of a Hausdorff space. (Contributed by NM, 8-Mar-2007.)
Hypothesis
Ref Expression
ist0.1
Assertion
Ref Expression
hausnei
Distinct variable groups:   ,,   ,,   ,,
Allowed substitution hints:   (,)

Proof of Theorem hausnei
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ist0.1 . . . . . . 7
21ishaus 17424 . . . . . 6
32simprbi 452 . . . . 5
4 neeq1 2616 . . . . . . 7
5 eleq1 2503 . . . . . . . . 9
653anbi1d 1259 . . . . . . . 8
762rexbidv 2755 . . . . . . 7
84, 7imbi12d 313 . . . . . 6
9 neeq2 2617 . . . . . . 7
10 eleq1 2503 . . . . . . . . 9
11103anbi2d 1260 . . . . . . . 8
12112rexbidv 2755 . . . . . . 7
139, 12imbi12d 313 . . . . . 6
148, 13rspc2v 3067 . . . . 5
153, 14syl5 31 . . . 4
1615ex 425 . . 3
1716com3r 76 . 2
18173imp2 1169 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   w3a 937   wceq 1654   wcel 1728   wne 2606  wral 2712  wrex 2713   cin 3308  c0 3616  cuni 4044  ctop 16996  cha 17410 This theorem is referenced by:  haust1  17454  cnhaus  17456  lmmo  17482  hauscmplem  17507  pthaus  17708  txhaus  17717  xkohaus  17723  hausflimi  18050  hauspwpwf1  18057 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 1628  ax-9 1669  ax-8 1690  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-rab 2721  df-v 2967  df-uni 4045  df-haus 17417
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