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Theorem pconcn 24911
 Description: The property of being a path-connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.)
Hypothesis
Ref Expression
ispcon.1
Assertion
Ref Expression
pconcn PCon
Distinct variable groups:   ,   ,   ,
Allowed substitution hint:   ()

Proof of Theorem pconcn
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ispcon.1 . . . . 5
21ispcon 24910 . . . 4 PCon
32simprbi 451 . . 3 PCon
4 eqeq2 2445 . . . . . 6
54anbi1d 686 . . . . 5
65rexbidv 2726 . . . 4
7 eqeq2 2445 . . . . . 6
87anbi2d 685 . . . . 5
98rexbidv 2726 . . . 4
106, 9rspc2v 3058 . . 3
113, 10syl5com 28 . 2 PCon
12113impib 1151 1 PCon
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   w3a 936   wceq 1652   wcel 1725  wral 2705  wrex 2706  cuni 4015  cfv 5454  (class class class)co 6081  cc0 8990  c1 8991  ctop 16958   ccn 17288  cii 18905  PConcpcon 24906 This theorem is referenced by:  cnpcon  24917  pconcon  24918  txpcon  24919  ptpcon  24920  conpcon  24922  pconpi1  24924  cvmlift3lem2  25007  cvmlift3lem7  25012 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 2417 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 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-ov 6084  df-pcon 24908
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