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Theorem isfrgra 25117
 Description: The property of being a friendship graph. (Contributed by Alexander van der Vekens, 4-Oct-2017.)
Assertion
Ref Expression
isfrgra FriendGrph USGrph
Distinct variable groups:   ,,,   ,,,
Allowed substitution hints:   (,,)   (,,)

Proof of Theorem isfrgra
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq1 4459 . . 3 USGrph USGrph
2 difeq1 3611 . . . . 5
3 reueq1 3056 . . . . 5
42, 3raleqbidv 3068 . . . 4
54raleqbi1dv 3062 . . 3
61, 5anbi12d 710 . 2 USGrph USGrph
7 breq2 4460 . . 3 USGrph USGrph
8 rneq 5238 . . . . . 6
98sseq2d 3527 . . . . 5
109reubidv 3042 . . . 4
11102ralbidv 2901 . . 3
127, 11anbi12d 710 . 2 USGrph USGrph
13 df-frgra 25116 . 2 FriendGrph USGrph
146, 12, 13brabg 4775 1 FriendGrph USGrph
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wceq 1395   wcel 1819  wral 2807  wreu 2809   cdif 3468   wss 3471  csn 4032  cpr 4034   class class class wbr 4456   crn 5009   USGrph cusg 24457   FriendGrph cfrgra 25115 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-reu 2814  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-cnv 5016  df-dm 5018  df-rn 5019  df-frgra 25116 This theorem is referenced by:  frisusgrapr  25118  frgra0v  25120  frgra1v  25125  frgra2v  25126  frgra3v  25129
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