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Theorem ishaus 17378
 Description: Express the predicate " is a Hausdorff space." (Contributed by NM, 8-Mar-2007.)
Hypothesis
Ref Expression
ist0.1
Assertion
Ref Expression
ishaus
Distinct variable groups:   ,   ,,,,   ,,
Allowed substitution hints:   (,)

Proof of Theorem ishaus
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 unieq 4016 . . . 4
2 ist0.1 . . . 4
31, 2syl6eqr 2485 . . 3
4 rexeq 2897 . . . . . 6
54rexeqbi1dv 2905 . . . . 5
65imbi2d 308 . . . 4
73, 6raleqbidv 2908 . . 3
83, 7raleqbidv 2908 . 2
9 df-haus 17371 . 2
108, 9elrab2 3086 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   w3a 936   wceq 1652   wcel 1725   wne 2598  wral 2697  wrex 2698   cin 3311  c0 3620  cuni 4007  ctop 16950  cha 17364 This theorem is referenced by:  hausnei  17384  haustop  17387  ishaus2  17407  cnhaus  17410  dishaus  17438  pthaus  17662  hausdiag  17669  txhaus  17671  xkohaus  17677 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-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-uni 4008  df-haus 17371
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