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Theorem frgr2wwlkeqm 29573
Description: If there is a (simple) path of length 2 from one vertex to another vertex and a (simple) path of length 2 from the other vertex back to the first vertex in a friendship graph, then the middle vertex is the same. This is only an observation, which is not required to proof the friendship theorem. (Contributed by Alexander van der Vekens, 20-Feb-2018.) (Revised by AV, 13-May-2021.) (Proof shortened by AV, 7-Jan-2022.)
Assertion
Ref Expression
frgr2wwlkeqm ((𝐺 ∈ FriendGraph ∧ 𝐎 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) → ((⟚“𝐎𝑃𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵) ∧ ⟚“𝐵𝑄𝐎”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐎)) → 𝑄 = 𝑃))

Proof of Theorem frgr2wwlkeqm
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simp3l 1201 . . . 4 ((𝐺 ∈ FriendGraph ∧ 𝐎 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) → 𝑃 ∈ 𝑋)
2 eqid 2732 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
32wwlks2onv 29196 . . . 4 ((𝑃 ∈ 𝑋 ∧ ⟚“𝐎𝑃𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵)) → (𝐎 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)))
41, 3sylan 580 . . 3 (((𝐺 ∈ FriendGraph ∧ 𝐎 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) ∧ ⟚“𝐎𝑃𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵)) → (𝐎 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)))
5 simp3r 1202 . . . . . . . 8 ((𝐺 ∈ FriendGraph ∧ 𝐎 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) → 𝑄 ∈ 𝑌)
62wwlks2onv 29196 . . . . . . . 8 ((𝑄 ∈ 𝑌 ∧ ⟚“𝐵𝑄𝐎”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐎)) → (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐎 ∈ (Vtx‘𝐺)))
75, 6sylan 580 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ 𝐎 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) ∧ ⟚“𝐵𝑄𝐎”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐎)) → (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐎 ∈ (Vtx‘𝐺)))
8 frgrusgr 29503 . . . . . . . . . . . . . . 15 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
9 usgrumgr 28428 . . . . . . . . . . . . . . 15 (𝐺 ∈ USGraph → 𝐺 ∈ UMGraph)
108, 9syl 17 . . . . . . . . . . . . . 14 (𝐺 ∈ FriendGraph → 𝐺 ∈ UMGraph)
11103ad2ant1 1133 . . . . . . . . . . . . 13 ((𝐺 ∈ FriendGraph ∧ 𝐎 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) → 𝐺 ∈ UMGraph)
12 simpr3 1196 . . . . . . . . . . . . . 14 ((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐎 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) → 𝐵 ∈ (Vtx‘𝐺))
13 simpl 483 . . . . . . . . . . . . . 14 ((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐎 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) → 𝑄 ∈ (Vtx‘𝐺))
14 simpr1 1194 . . . . . . . . . . . . . 14 ((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐎 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) → 𝐎 ∈ (Vtx‘𝐺))
1512, 13, 143jca 1128 . . . . . . . . . . . . 13 ((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐎 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) → (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐎 ∈ (Vtx‘𝐺)))
162wwlks2onsym 29201 . . . . . . . . . . . . 13 ((𝐺 ∈ UMGraph ∧ (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐎 ∈ (Vtx‘𝐺))) → (⟚“𝐵𝑄𝐎”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐎) ↔ ⟚“𝐎𝑄𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵)))
1711, 15, 16syl2anr 597 . . . . . . . . . . . 12 (((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐎 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐎 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌))) → (⟚“𝐵𝑄𝐎”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐎) ↔ ⟚“𝐎𝑄𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵)))
18 simpr1 1194 . . . . . . . . . . . . . 14 (((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐎 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐎 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌))) → 𝐺 ∈ FriendGraph )
19 3simpb 1149 . . . . . . . . . . . . . . 15 ((𝐎 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → (𝐎 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)))
2019ad2antlr 725 . . . . . . . . . . . . . 14 (((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐎 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐎 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌))) → (𝐎 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)))
21 simpr2 1195 . . . . . . . . . . . . . 14 (((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐎 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐎 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌))) → 𝐎 ≠ 𝐵)
222frgr2wwlkeu 29569 . . . . . . . . . . . . . 14 ((𝐺 ∈ FriendGraph ∧ (𝐎 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ 𝐎 ≠ 𝐵) → ∃!𝑥 ∈ (Vtx‘𝐺)⟚“𝐎𝑥𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵))
2318, 20, 21, 22syl3anc 1371 . . . . . . . . . . . . 13 (((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐎 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐎 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌))) → ∃!𝑥 ∈ (Vtx‘𝐺)⟚“𝐎𝑥𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵))
24 s3eq2 14817 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑄 → ⟚“𝐎𝑥𝐵”⟩ = ⟚“𝐎𝑄𝐵”⟩)
2524eleq1d 2818 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑄 → (⟚“𝐎𝑥𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵) ↔ ⟚“𝐎𝑄𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵)))
2625riota2 7387 . . . . . . . . . . . . . . 15 ((𝑄 ∈ (Vtx‘𝐺) ∧ ∃!𝑥 ∈ (Vtx‘𝐺)⟚“𝐎𝑥𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵)) → (⟚“𝐎𝑄𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵) ↔ (℩𝑥 ∈ (Vtx‘𝐺)⟚“𝐎𝑥𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵)) = 𝑄))
2726ad4ant14 750 . . . . . . . . . . . . . 14 ((((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐎 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐎 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌))) ∧ ∃!𝑥 ∈ (Vtx‘𝐺)⟚“𝐎𝑥𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵)) → (⟚“𝐎𝑄𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵) ↔ (℩𝑥 ∈ (Vtx‘𝐺)⟚“𝐎𝑥𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵)) = 𝑄))
28 simplr2 1216 . . . . . . . . . . . . . . . . 17 (((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐎 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐎 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌))) → 𝑃 ∈ (Vtx‘𝐺))
29 s3eq2 14817 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑃 → ⟚“𝐎𝑥𝐵”⟩ = ⟚“𝐎𝑃𝐵”⟩)
3029eleq1d 2818 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑃 → (⟚“𝐎𝑥𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵) ↔ ⟚“𝐎𝑃𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵)))
3130riota2 7387 . . . . . . . . . . . . . . . . 17 ((𝑃 ∈ (Vtx‘𝐺) ∧ ∃!𝑥 ∈ (Vtx‘𝐺)⟚“𝐎𝑥𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵)) → (⟚“𝐎𝑃𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵) ↔ (℩𝑥 ∈ (Vtx‘𝐺)⟚“𝐎𝑥𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵)) = 𝑃))
3228, 31sylan 580 . . . . . . . . . . . . . . . 16 ((((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐎 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐎 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌))) ∧ ∃!𝑥 ∈ (Vtx‘𝐺)⟚“𝐎𝑥𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵)) → (⟚“𝐎𝑃𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵) ↔ (℩𝑥 ∈ (Vtx‘𝐺)⟚“𝐎𝑥𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵)) = 𝑃))
33 eqtr2 2756 . . . . . . . . . . . . . . . . 17 (((℩𝑥 ∈ (Vtx‘𝐺)⟚“𝐎𝑥𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵)) = 𝑄 ∧ (℩𝑥 ∈ (Vtx‘𝐺)⟚“𝐎𝑥𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵)) = 𝑃) → 𝑄 = 𝑃)
3433expcom 414 . . . . . . . . . . . . . . . 16 ((℩𝑥 ∈ (Vtx‘𝐺)⟚“𝐎𝑥𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵)) = 𝑃 → ((℩𝑥 ∈ (Vtx‘𝐺)⟚“𝐎𝑥𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵)) = 𝑄 → 𝑄 = 𝑃))
3532, 34syl6bi 252 . . . . . . . . . . . . . . 15 ((((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐎 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐎 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌))) ∧ ∃!𝑥 ∈ (Vtx‘𝐺)⟚“𝐎𝑥𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵)) → (⟚“𝐎𝑃𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵) → ((℩𝑥 ∈ (Vtx‘𝐺)⟚“𝐎𝑥𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵)) = 𝑄 → 𝑄 = 𝑃)))
3635com23 86 . . . . . . . . . . . . . 14 ((((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐎 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐎 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌))) ∧ ∃!𝑥 ∈ (Vtx‘𝐺)⟚“𝐎𝑥𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵)) → ((℩𝑥 ∈ (Vtx‘𝐺)⟚“𝐎𝑥𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵)) = 𝑄 → (⟚“𝐎𝑃𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵) → 𝑄 = 𝑃)))
3727, 36sylbid 239 . . . . . . . . . . . . 13 ((((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐎 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐎 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌))) ∧ ∃!𝑥 ∈ (Vtx‘𝐺)⟚“𝐎𝑥𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵)) → (⟚“𝐎𝑄𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵) → (⟚“𝐎𝑃𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵) → 𝑄 = 𝑃)))
3823, 37mpdan 685 . . . . . . . . . . . 12 (((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐎 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐎 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌))) → (⟚“𝐎𝑄𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵) → (⟚“𝐎𝑃𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵) → 𝑄 = 𝑃)))
3917, 38sylbid 239 . . . . . . . . . . 11 (((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐎 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐎 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌))) → (⟚“𝐵𝑄𝐎”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐎) → (⟚“𝐎𝑃𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵) → 𝑄 = 𝑃)))
4039expimpd 454 . . . . . . . . . 10 ((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐎 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) → (((𝐺 ∈ FriendGraph ∧ 𝐎 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) ∧ ⟚“𝐵𝑄𝐎”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐎)) → (⟚“𝐎𝑃𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵) → 𝑄 = 𝑃)))
4140ex 413 . . . . . . . . 9 (𝑄 ∈ (Vtx‘𝐺) → ((𝐎 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → (((𝐺 ∈ FriendGraph ∧ 𝐎 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) ∧ ⟚“𝐵𝑄𝐎”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐎)) → (⟚“𝐎𝑃𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵) → 𝑄 = 𝑃))))
4241com23 86 . . . . . . . 8 (𝑄 ∈ (Vtx‘𝐺) → (((𝐺 ∈ FriendGraph ∧ 𝐎 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) ∧ ⟚“𝐵𝑄𝐎”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐎)) → ((𝐎 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → (⟚“𝐎𝑃𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵) → 𝑄 = 𝑃))))
43423ad2ant2 1134 . . . . . . 7 ((𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐎 ∈ (Vtx‘𝐺)) → (((𝐺 ∈ FriendGraph ∧ 𝐎 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) ∧ ⟚“𝐵𝑄𝐎”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐎)) → ((𝐎 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → (⟚“𝐎𝑃𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵) → 𝑄 = 𝑃))))
447, 43mpcom 38 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ 𝐎 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) ∧ ⟚“𝐵𝑄𝐎”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐎)) → ((𝐎 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → (⟚“𝐎𝑃𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵) → 𝑄 = 𝑃)))
4544ex 413 . . . . 5 ((𝐺 ∈ FriendGraph ∧ 𝐎 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) → (⟚“𝐵𝑄𝐎”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐎) → ((𝐎 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → (⟚“𝐎𝑃𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵) → 𝑄 = 𝑃))))
4645com24 95 . . . 4 ((𝐺 ∈ FriendGraph ∧ 𝐎 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) → (⟚“𝐎𝑃𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵) → ((𝐎 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → (⟚“𝐵𝑄𝐎”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐎) → 𝑄 = 𝑃))))
4746imp 407 . . 3 (((𝐺 ∈ FriendGraph ∧ 𝐎 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) ∧ ⟚“𝐎𝑃𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵)) → ((𝐎 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → (⟚“𝐵𝑄𝐎”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐎) → 𝑄 = 𝑃)))
484, 47mpd 15 . 2 (((𝐺 ∈ FriendGraph ∧ 𝐎 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) ∧ ⟚“𝐎𝑃𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵)) → (⟚“𝐵𝑄𝐎”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐎) → 𝑄 = 𝑃))
4948expimpd 454 1 ((𝐺 ∈ FriendGraph ∧ 𝐎 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) → ((⟚“𝐎𝑃𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵) ∧ ⟚“𝐵𝑄𝐎”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐎)) → 𝑄 = 𝑃))
Colors of variables: wff setvar class
Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2940  âˆƒ!wreu 3374  â€˜cfv 6540  â„©crio 7360  (class class class)co 7405  2c2 12263  âŸšâ€œcs3 14789  Vtxcvtx 28245  UMGraphcumgr 28330  USGraphcusgr 28398   WWalksNOn cwwlksnon 29070   FriendGraph cfrgr 29500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-ac2 10454  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ifp 1062  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-oadd 8466  df-er 8699  df-map 8818  df-pm 8819  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-dju 9892  df-card 9930  df-ac 10107  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-n0 12469  df-xnn0 12541  df-z 12555  df-uz 12819  df-fz 13481  df-fzo 13624  df-hash 14287  df-word 14461  df-concat 14517  df-s1 14542  df-s2 14795  df-s3 14796  df-edg 28297  df-uhgr 28307  df-upgr 28331  df-umgr 28332  df-usgr 28400  df-wlks 28845  df-wwlks 29073  df-wwlksn 29074  df-wwlksnon 29075  df-frgr 29501
This theorem is referenced by: (None)
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