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Theorem frgr2wwlkeqm 29317
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 1202 . . . 4 ((𝐺 ∈ FriendGraph ∧ 𝐎 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) → 𝑃 ∈ 𝑋)
2 eqid 2737 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
32wwlks2onv 28940 . . . 4 ((𝑃 ∈ 𝑋 ∧ ⟚“𝐎𝑃𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵)) → (𝐎 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)))
41, 3sylan 581 . . 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 1203 . . . . . . . 8 ((𝐺 ∈ FriendGraph ∧ 𝐎 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) → 𝑄 ∈ 𝑌)
62wwlks2onv 28940 . . . . . . . 8 ((𝑄 ∈ 𝑌 ∧ ⟚“𝐵𝑄𝐎”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐎)) → (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐎 ∈ (Vtx‘𝐺)))
75, 6sylan 581 . . . . . . 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 29247 . . . . . . . . . . . . . . 15 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
9 usgrumgr 28172 . . . . . . . . . . . . . . 15 (𝐺 ∈ USGraph → 𝐺 ∈ UMGraph)
108, 9syl 17 . . . . . . . . . . . . . 14 (𝐺 ∈ FriendGraph → 𝐺 ∈ UMGraph)
11103ad2ant1 1134 . . . . . . . . . . . . 13 ((𝐺 ∈ FriendGraph ∧ 𝐎 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) → 𝐺 ∈ UMGraph)
12 simpr3 1197 . . . . . . . . . . . . . 14 ((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐎 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) → 𝐵 ∈ (Vtx‘𝐺))
13 simpl 484 . . . . . . . . . . . . . 14 ((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐎 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) → 𝑄 ∈ (Vtx‘𝐺))
14 simpr1 1195 . . . . . . . . . . . . . 14 ((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐎 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) → 𝐎 ∈ (Vtx‘𝐺))
1512, 13, 143jca 1129 . . . . . . . . . . . . 13 ((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐎 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) → (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐎 ∈ (Vtx‘𝐺)))
162wwlks2onsym 28945 . . . . . . . . . . . . 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 598 . . . . . . . . . . . 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 1195 . . . . . . . . . . . . . 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 1150 . . . . . . . . . . . . . . 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 1196 . . . . . . . . . . . . . 14 (((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐎 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐎 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌))) → 𝐎 ≠ 𝐵)
222frgr2wwlkeu 29313 . . . . . . . . . . . . . 14 ((𝐺 ∈ FriendGraph ∧ (𝐎 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ 𝐎 ≠ 𝐵) → ∃!𝑥 ∈ (Vtx‘𝐺)⟚“𝐎𝑥𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵))
2318, 20, 21, 22syl3anc 1372 . . . . . . . . . . . . 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 14766 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑄 → ⟚“𝐎𝑥𝐵”⟩ = ⟚“𝐎𝑄𝐵”⟩)
2524eleq1d 2823 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑄 → (⟚“𝐎𝑥𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵) ↔ ⟚“𝐎𝑄𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵)))
2625riota2 7344 . . . . . . . . . . . . . . 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 1217 . . . . . . . . . . . . . . . . 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 14766 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑃 → ⟚“𝐎𝑥𝐵”⟩ = ⟚“𝐎𝑃𝐵”⟩)
3029eleq1d 2823 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑃 → (⟚“𝐎𝑥𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵) ↔ ⟚“𝐎𝑃𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵)))
3130riota2 7344 . . . . . . . . . . . . . . . . 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 581 . . . . . . . . . . . . . . . 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 2761 . . . . . . . . . . . . . . . . 17 (((℩𝑥 ∈ (Vtx‘𝐺)⟚“𝐎𝑥𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵)) = 𝑄 ∧ (℩𝑥 ∈ (Vtx‘𝐺)⟚“𝐎𝑥𝐵”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐵)) = 𝑃) → 𝑄 = 𝑃)
3433expcom 415 . . . . . . . . . . . . . . . 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 253 . . . . . . . . . . . . . . 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 455 . . . . . . . . . 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 414 . . . . . . . . 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 1135 . . . . . . 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 414 . . . . 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 408 . . 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 455 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   ≠ wne 2944  âˆƒ!wreu 3354  â€˜cfv 6501  â„©crio 7317  (class class class)co 7362  2c2 12215  âŸšâ€œcs3 14738  Vtxcvtx 27989  UMGraphcumgr 28074  USGraphcusgr 28142   WWalksNOn cwwlksnon 28814   FriendGraph cfrgr 29244
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-ac2 10406  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ifp 1063  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-2o 8418  df-oadd 8421  df-er 8655  df-map 8774  df-pm 8775  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-dju 9844  df-card 9882  df-ac 10059  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-nn 12161  df-2 12223  df-3 12224  df-n0 12421  df-xnn0 12493  df-z 12507  df-uz 12771  df-fz 13432  df-fzo 13575  df-hash 14238  df-word 14410  df-concat 14466  df-s1 14491  df-s2 14744  df-s3 14745  df-edg 28041  df-uhgr 28051  df-upgr 28075  df-umgr 28076  df-usgr 28144  df-wlks 28589  df-wwlks 28817  df-wwlksn 28818  df-wwlksnon 28819  df-frgr 29245
This theorem is referenced by: (None)
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