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Theorem frcond2 41439
 Description: The friendship condition: any two (different) vertices in a friendship graph have a unique common neighbor. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 29-Mar-2021.)
Hypotheses
Ref Expression
frcond1.v 𝑉 = (Vtx‘𝐺)
frcond1.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
frcond2 (𝐺 ∈ FriendGraph → ((𝐴𝑉𝐶𝑉𝐴𝐶) → ∃!𝑏𝑉 ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸)))
Distinct variable groups:   𝐴,𝑏   𝐶,𝑏   𝐺,𝑏   𝑉,𝑏
Allowed substitution hint:   𝐸(𝑏)

Proof of Theorem frcond2
StepHypRef Expression
1 frcond1.v . . 3 𝑉 = (Vtx‘𝐺)
2 frcond1.e . . 3 𝐸 = (Edg‘𝐺)
31, 2frcond1 41438 . 2 (𝐺 ∈ FriendGraph → ((𝐴𝑉𝐶𝑉𝐴𝐶) → ∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸))
4 prex 4836 . . . . 5 {𝐴, 𝑏} ∈ V
5 prex 4836 . . . . 5 {𝑏, 𝐶} ∈ V
64, 5prss 4291 . . . 4 (({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸) ↔ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸)
76bicomi 213 . . 3 ({{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸 ↔ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸))
87reubii 3105 . 2 (∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸 ↔ ∃!𝑏𝑉 ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸))
93, 8syl6ib 240 1 (𝐺 ∈ FriendGraph → ((𝐴𝑉𝐶𝑉𝐴𝐶) → ∃!𝑏𝑉 ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃!wreu 2898   ⊆ wss 3540  {cpr 4127  ‘cfv 5804  Vtxcvtx 25673  Edgcedga 25792   FriendGraph cfrgr 41428 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-frgr 41429 This theorem is referenced by:  frgrncvvdeqlemC  41478  frgreu  41491  frgr2wwlkeu  41492  frgr2wwlkeqm  41496  av-numclwwlk2lem1  41532
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