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Theorem regsep 17390
 Description: In a regular space, every neighborhood of a point contains a closed subneighborhood. (Contributed by Mario Carneiro, 25-Aug-2015.)
Assertion
Ref Expression
regsep
Distinct variable groups:   ,   ,   ,

Proof of Theorem regsep
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isreg 17388 . . . . 5
21simprbi 451 . . . 4
3 sseq2 3362 . . . . . . . 8
43anbi2d 685 . . . . . . 7
54rexbidv 2718 . . . . . 6
65raleqbi1dv 2904 . . . . 5
76rspccv 3041 . . . 4
82, 7syl 16 . . 3
9 eleq1 2495 . . . . . 6
109anbi1d 686 . . . . 5
1110rexbidv 2718 . . . 4
1211rspccv 3041 . . 3
138, 12syl6 31 . 2
14133imp 1147 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   w3a 936   wceq 1652   wcel 1725  wral 2697  wrex 2698   wss 3312  cfv 5446  ctop 16950  ccl 17074  creg 17365 This theorem is referenced by:  regsep2  17432  regr1lem  17763  kqreglem1  17765  kqreglem2  17766  reghmph  17817  cnextcn  18090 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-3an 938  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-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-reg 17372
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