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Theorem frgr2wwlkeqm 30128
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 1199 . . . 4 ((𝐺 ∈ FriendGraph ∧ 𝐎 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) → 𝑃 ∈ 𝑋)
2 eqid 2727 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
32wwlks2onv 29751 . . . 4 ((𝑃 ∈ 𝑋 ∧ ⟚“𝐎𝑃𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵)) → (𝐎 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)))
41, 3sylan 579 . . 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 1200 . . . . . . . 8 ((𝐺 ∈ FriendGraph ∧ 𝐎 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) → 𝑄 ∈ 𝑌)
62wwlks2onv 29751 . . . . . . . 8 ((𝑄 ∈ 𝑌 ∧ ⟚“𝐵𝑄𝐎”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐎)) → (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐎 ∈ (Vtx‘𝐺)))
75, 6sylan 579 . . . . . . 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 30058 . . . . . . . . . . . . . . 15 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
9 usgrumgr 28981 . . . . . . . . . . . . . . 15 (𝐺 ∈ USGraph → 𝐺 ∈ UMGraph)
108, 9syl 17 . . . . . . . . . . . . . 14 (𝐺 ∈ FriendGraph → 𝐺 ∈ UMGraph)
11103ad2ant1 1131 . . . . . . . . . . . . 13 ((𝐺 ∈ FriendGraph ∧ 𝐎 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) → 𝐺 ∈ UMGraph)
12 simpr3 1194 . . . . . . . . . . . . . 14 ((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐎 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) → 𝐵 ∈ (Vtx‘𝐺))
13 simpl 482 . . . . . . . . . . . . . 14 ((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐎 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) → 𝑄 ∈ (Vtx‘𝐺))
14 simpr1 1192 . . . . . . . . . . . . . 14 ((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐎 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) → 𝐎 ∈ (Vtx‘𝐺))
1512, 13, 143jca 1126 . . . . . . . . . . . . 13 ((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐎 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) → (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐎 ∈ (Vtx‘𝐺)))
162wwlks2onsym 29756 . . . . . . . . . . . . 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 596 . . . . . . . . . . . 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 1192 . . . . . . . . . . . . . 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 1147 . . . . . . . . . . . . . . 15 ((𝐎 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → (𝐎 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)))
2019ad2antlr 726 . . . . . . . . . . . . . 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 1193 . . . . . . . . . . . . . 14 (((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐎 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐎 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌))) → 𝐎 ≠ 𝐵)
222frgr2wwlkeu 30124 . . . . . . . . . . . . . 14 ((𝐺 ∈ FriendGraph ∧ (𝐎 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ 𝐎 ≠ 𝐵) → ∃!𝑥 ∈ (Vtx‘𝐺)⟚“𝐎𝑥𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵))
2318, 20, 21, 22syl3anc 1369 . . . . . . . . . . . . 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 14845 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑄 → ⟚“𝐎𝑥𝐵”⟩ = ⟚“𝐎𝑄𝐵”⟩)
2524eleq1d 2813 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑄 → (⟚“𝐎𝑥𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵) ↔ ⟚“𝐎𝑄𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵)))
2625riota2 7396 . . . . . . . . . . . . . . 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 751 . . . . . . . . . . . . . 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 1214 . . . . . . . . . . . . . . . . 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 14845 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑃 → ⟚“𝐎𝑥𝐵”⟩ = ⟚“𝐎𝑃𝐵”⟩)
3029eleq1d 2813 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑃 → (⟚“𝐎𝑥𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵) ↔ ⟚“𝐎𝑃𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵)))
3130riota2 7396 . . . . . . . . . . . . . . . . 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 579 . . . . . . . . . . . . . . . 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 2751 . . . . . . . . . . . . . . . . 17 (((℩𝑥 ∈ (Vtx‘𝐺)⟚“𝐎𝑥𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵)) = 𝑄 ∧ (℩𝑥 ∈ (Vtx‘𝐺)⟚“𝐎𝑥𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵)) = 𝑃) → 𝑄 = 𝑃)
3433expcom 413 . . . . . . . . . . . . . . . 16 ((℩𝑥 ∈ (Vtx‘𝐺)⟚“𝐎𝑥𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵)) = 𝑃 → ((℩𝑥 ∈ (Vtx‘𝐺)⟚“𝐎𝑥𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵)) = 𝑄 → 𝑄 = 𝑃))
3532, 34biimtrdi 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 686 . . . . . . . . . . . 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 453 . . . . . . . . . 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 412 . . . . . . . . 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 1132 . . . . . . 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 412 . . . . 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 406 . . 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 453 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 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   ≠ wne 2935  âˆƒ!wreu 3369  â€˜cfv 6542  â„©crio 7369  (class class class)co 7414  2c2 12289  âŸšâ€œcs3 14817  Vtxcvtx 28796  UMGraphcumgr 28881  USGraphcusgr 28949   WWalksNOn cwwlksnon 29625   FriendGraph cfrgr 30055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-ac2 10478  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-ifp 1062  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-oadd 8484  df-er 8718  df-map 8838  df-pm 8839  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-dju 9916  df-card 9954  df-ac 10131  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-nn 12235  df-2 12297  df-3 12298  df-n0 12495  df-xnn0 12567  df-z 12581  df-uz 12845  df-fz 13509  df-fzo 13652  df-hash 14314  df-word 14489  df-concat 14545  df-s1 14570  df-s2 14823  df-s3 14824  df-edg 28848  df-uhgr 28858  df-upgr 28882  df-umgr 28883  df-usgr 28951  df-wlks 29400  df-wwlks 29628  df-wwlksn 29629  df-wwlksnon 29630  df-frgr 30056
This theorem is referenced by: (None)
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