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Theorem nbgrasym 25012
 Description: A vertex in a graph is a neighbor of a second vertex if and only if the second vertex is a neighbor of the first vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
Assertion
Ref Expression
nbgrasym USGrph Neighbors Neighbors

Proof of Theorem nbgrasym
StepHypRef Expression
1 usgrav 24911 . 2 USGrph
2 3ancoma 989 . . . . 5
3 prcom 4081 . . . . . . 7
43eleq1i 2506 . . . . . 6
543anbi3i 1198 . . . . 5
62, 5bitri 252 . . . 4
76a1i 11 . . 3
8 nbgrael 24999 . . 3 Neighbors
9 nbgrael 24999 . . 3 Neighbors
107, 8, 93bitr4d 288 . 2 Neighbors Neighbors
111, 10syl 17 1 USGrph Neighbors Neighbors
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 187   wa 370   w3a 982   wcel 1870  cvv 3087  cpr 4004  cop 4008   class class class wbr 4426   crn 4855  (class class class)co 6305   USGrph cusg 24903   Neighbors cnbgra 24990 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-usgra 24906  df-nbgra 24993 This theorem is referenced by:  uvtxnbgravtx  25068
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