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Theorem t0sep 17380
 Description: Any two topologically indistinguishable points in a T0 space are identical. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
ist0.1
Assertion
Ref Expression
t0sep
Distinct variable groups:   ,   ,   ,   ,

Proof of Theorem t0sep
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ist0.1 . . . 4
21ist0 17376 . . 3
32simprbi 451 . 2
4 eleq1 2495 . . . . . . 7
54bibi1d 311 . . . . . 6
65ralbidv 2717 . . . . 5
7 eqeq1 2441 . . . . 5
86, 7imbi12d 312 . . . 4
9 eleq1 2495 . . . . . . 7
109bibi2d 310 . . . . . 6
1110ralbidv 2717 . . . . 5
12 eqeq2 2444 . . . . 5
1311, 12imbi12d 312 . . . 4
148, 13rspc2va 3051 . . 3
1514ancoms 440 . 2
163, 15sylan 458 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  wral 2697  cuni 4007  ctop 16950  ct0 17362 This theorem is referenced by:  t0dist  17381  cnt0  17402 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-t0 17369
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