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Theorem frg2woteq 26587
 Description: There is a (simple) path of length 2 from one vertex to another vertex in a friendship graph if and only if there is a (simple) path of length 2 from the other vertex back to the first vertex. (Contributed by Alexander van der Vekens, 14-Feb-2018.)
Assertion
Ref Expression
frg2woteq ((𝑉 FriendGrph 𝐸𝐴𝐵) → ((𝑃 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ 𝑄 ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴)) → ((1st ‘(1st𝑃)) = (2nd𝑄) ∧ (2nd ‘(1st𝑃)) = (2nd ‘(1st𝑃)) ∧ (1st ‘(1st𝑄)) = (2nd𝑃))))

Proof of Theorem frg2woteq
Dummy variables 𝑐 𝑑 𝑓 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 2wlkonot3v 26402 . . . 4 (𝑃 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)))
21adantr 480 . . 3 ((𝑃 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ 𝑄 ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴)) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)))
3 el2wlkonot 26396 . . . . . 6 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → (𝑃 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ↔ ∃𝑐𝑉 (𝑃 = ⟨𝐴, 𝑐, 𝐵⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑐 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2))))))
4 pm3.22 464 . . . . . . . 8 ((𝐴𝑉𝐵𝑉) → (𝐵𝑉𝐴𝑉))
54anim2i 591 . . . . . . 7 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐵𝑉𝐴𝑉)))
6 el2wlkonot 26396 . . . . . . 7 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐵𝑉𝐴𝑉)) → (𝑄 ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴) ↔ ∃𝑑𝑉 (𝑄 = ⟨𝐵, 𝑑, 𝐴⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐵 = (𝑝‘0) ∧ 𝑑 = (𝑝‘1) ∧ 𝐴 = (𝑝‘2))))))
75, 6syl 17 . . . . . 6 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → (𝑄 ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴) ↔ ∃𝑑𝑉 (𝑄 = ⟨𝐵, 𝑑, 𝐴⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐵 = (𝑝‘0) ∧ 𝑑 = (𝑝‘1) ∧ 𝐴 = (𝑝‘2))))))
83, 7anbi12d 743 . . . . 5 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → ((𝑃 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ 𝑄 ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴)) ↔ (∃𝑐𝑉 (𝑃 = ⟨𝐴, 𝑐, 𝐵⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑐 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2)))) ∧ ∃𝑑𝑉 (𝑄 = ⟨𝐵, 𝑑, 𝐴⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐵 = (𝑝‘0) ∧ 𝑑 = (𝑝‘1) ∧ 𝐴 = (𝑝‘2)))))))
983adant3 1074 . . . 4 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)) → ((𝑃 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ 𝑄 ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴)) ↔ (∃𝑐𝑉 (𝑃 = ⟨𝐴, 𝑐, 𝐵⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑐 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2)))) ∧ ∃𝑑𝑉 (𝑄 = ⟨𝐵, 𝑑, 𝐴⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐵 = (𝑝‘0) ∧ 𝑑 = (𝑝‘1) ∧ 𝐴 = (𝑝‘2)))))))
1053adant3 1074 . . . . . . . . . . . . 13 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐵𝑉𝐴𝑉)))
1110adantr 480 . . . . . . . . . . . 12 ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)) ∧ 𝑐𝑉) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐵𝑉𝐴𝑉)))
1211ad2antrr 758 . . . . . . . . . . 11 ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)) ∧ 𝑐𝑉) ∧ 𝑑𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐵𝑉𝐴𝑉)))
13 el2wlkonotot 26400 . . . . . . . . . . . 12 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐵𝑉𝐴𝑉)) → (⟨𝐵, 𝑑, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴) ↔ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐵 = (𝑝‘0) ∧ 𝑑 = (𝑝‘1) ∧ 𝐴 = (𝑝‘2)))))
1413bicomd 212 . . . . . . . . . . 11 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐵𝑉𝐴𝑉)) → (∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐵 = (𝑝‘0) ∧ 𝑑 = (𝑝‘1) ∧ 𝐴 = (𝑝‘2))) ↔ ⟨𝐵, 𝑑, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴)))
1512, 14syl 17 . . . . . . . . . 10 ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)) ∧ 𝑐𝑉) ∧ 𝑑𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) → (∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐵 = (𝑝‘0) ∧ 𝑑 = (𝑝‘1) ∧ 𝐴 = (𝑝‘2))) ↔ ⟨𝐵, 𝑑, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴)))
16 3simpa 1051 . . . . . . . . . . . . . . . 16 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉)))
1716ad2antrr 758 . . . . . . . . . . . . . . 15 (((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)) ∧ 𝑐𝑉) ∧ 𝑑𝑉) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉)))
1817ad2antrr 758 . . . . . . . . . . . . . 14 (((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)) ∧ 𝑐𝑉) ∧ 𝑑𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉)))
19 el2wlkonotot 26400 . . . . . . . . . . . . . . 15 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → (⟨𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ↔ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑐 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2)))))
2019bicomd 212 . . . . . . . . . . . . . 14 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → (∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑐 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2))) ↔ ⟨𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)))
2118, 20syl 17 . . . . . . . . . . . . 13 (((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)) ∧ 𝑐𝑉) ∧ 𝑑𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩) → (∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑐 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2))) ↔ ⟨𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)))
22 frg2woteqm 26586 . . . . . . . . . . . . . . . . . . . . 21 ((𝑉 FriendGrph 𝐸𝐴𝐵) → ((⟨𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ⟨𝐵, 𝑑, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴)) → 𝑑 = 𝑐))
23 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑃 = ⟨𝐴, 𝑐, 𝐵⟩ → (1st𝑃) = (1st ‘⟨𝐴, 𝑐, 𝐵⟩))
2423fveq2d 6107 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑃 = ⟨𝐴, 𝑐, 𝐵⟩ → (1st ‘(1st𝑃)) = (1st ‘(1st ‘⟨𝐴, 𝑐, 𝐵⟩)))
2524adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐴𝑉𝐵𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩) → (1st ‘(1st𝑃)) = (1st ‘(1st ‘⟨𝐴, 𝑐, 𝐵⟩)))
2625adantl 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑑 = 𝑐 ∧ (((𝐴𝑉𝐵𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩)) → (1st ‘(1st𝑃)) = (1st ‘(1st ‘⟨𝐴, 𝑐, 𝐵⟩)))
27 vex 3176 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑐 ∈ V
28 ot1stg 7073 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐴𝑉𝑐 ∈ V ∧ 𝐵𝑉) → (1st ‘(1st ‘⟨𝐴, 𝑐, 𝐵⟩)) = 𝐴)
2927, 28mp3an2 1404 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐴𝑉𝐵𝑉) → (1st ‘(1st ‘⟨𝐴, 𝑐, 𝐵⟩)) = 𝐴)
3029ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐴𝑉𝐵𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩) → (1st ‘(1st ‘⟨𝐴, 𝑐, 𝐵⟩)) = 𝐴)
3130adantl 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑑 = 𝑐 ∧ (((𝐴𝑉𝐵𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩)) → (1st ‘(1st ‘⟨𝐴, 𝑐, 𝐵⟩)) = 𝐴)
32 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑄 = ⟨𝐵, 𝑑, 𝐴⟩ → (2nd𝑄) = (2nd ‘⟨𝐵, 𝑑, 𝐴⟩))
3332ad2antlr 759 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝐴𝑉𝐵𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩) → (2nd𝑄) = (2nd ‘⟨𝐵, 𝑑, 𝐴⟩))
3433adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑑 = 𝑐 ∧ (((𝐴𝑉𝐵𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩)) → (2nd𝑄) = (2nd ‘⟨𝐵, 𝑑, 𝐴⟩))
35 ot3rdg 7075 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐴𝑉 → (2nd ‘⟨𝐵, 𝑑, 𝐴⟩) = 𝐴)
3635adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐴𝑉𝐵𝑉) → (2nd ‘⟨𝐵, 𝑑, 𝐴⟩) = 𝐴)
3736ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝐴𝑉𝐵𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩) → (2nd ‘⟨𝐵, 𝑑, 𝐴⟩) = 𝐴)
3837adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑑 = 𝑐 ∧ (((𝐴𝑉𝐵𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩)) → (2nd ‘⟨𝐵, 𝑑, 𝐴⟩) = 𝐴)
3934, 38eqtr2d 2645 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑑 = 𝑐 ∧ (((𝐴𝑉𝐵𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩)) → 𝐴 = (2nd𝑄))
4026, 31, 393eqtrd 2648 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑑 = 𝑐 ∧ (((𝐴𝑉𝐵𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩)) → (1st ‘(1st𝑃)) = (2nd𝑄))
41 eqidd 2611 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑑 = 𝑐 ∧ (((𝐴𝑉𝐵𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩)) → (2nd ‘(1st𝑃)) = (2nd ‘(1st𝑃)))
42 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑄 = ⟨𝐵, 𝑑, 𝐴⟩ → (1st𝑄) = (1st ‘⟨𝐵, 𝑑, 𝐴⟩))
4342fveq2d 6107 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑄 = ⟨𝐵, 𝑑, 𝐴⟩ → (1st ‘(1st𝑄)) = (1st ‘(1st ‘⟨𝐵, 𝑑, 𝐴⟩)))
4443ad2antlr 759 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐴𝑉𝐵𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩) → (1st ‘(1st𝑄)) = (1st ‘(1st ‘⟨𝐵, 𝑑, 𝐴⟩)))
4544adantl 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑑 = 𝑐 ∧ (((𝐴𝑉𝐵𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩)) → (1st ‘(1st𝑄)) = (1st ‘(1st ‘⟨𝐵, 𝑑, 𝐴⟩)))
46 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐴𝑉𝐵𝑉) → 𝐵𝑉)
47 vex 3176 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 𝑑 ∈ V
4847a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐴𝑉𝐵𝑉) → 𝑑 ∈ V)
49 simpl 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐴𝑉𝐵𝑉) → 𝐴𝑉)
5046, 48, 493jca 1235 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐴𝑉𝐵𝑉) → (𝐵𝑉𝑑 ∈ V ∧ 𝐴𝑉))
5150ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝐴𝑉𝐵𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩) → (𝐵𝑉𝑑 ∈ V ∧ 𝐴𝑉))
5251adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑑 = 𝑐 ∧ (((𝐴𝑉𝐵𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩)) → (𝐵𝑉𝑑 ∈ V ∧ 𝐴𝑉))
53 ot1stg 7073 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐵𝑉𝑑 ∈ V ∧ 𝐴𝑉) → (1st ‘(1st ‘⟨𝐵, 𝑑, 𝐴⟩)) = 𝐵)
5452, 53syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑑 = 𝑐 ∧ (((𝐴𝑉𝐵𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩)) → (1st ‘(1st ‘⟨𝐵, 𝑑, 𝐴⟩)) = 𝐵)
55 ot3rdg 7075 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐵𝑉 → (2nd ‘⟨𝐴, 𝑐, 𝐵⟩) = 𝐵)
5655adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐴𝑉𝐵𝑉) → (2nd ‘⟨𝐴, 𝑐, 𝐵⟩) = 𝐵)
5756ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝐴𝑉𝐵𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩) → (2nd ‘⟨𝐴, 𝑐, 𝐵⟩) = 𝐵)
5857adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑑 = 𝑐 ∧ (((𝐴𝑉𝐵𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩)) → (2nd ‘⟨𝐴, 𝑐, 𝐵⟩) = 𝐵)
59 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝐴𝑉𝐵𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩) → 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩)
6059adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑑 = 𝑐 ∧ (((𝐴𝑉𝐵𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩)) → 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩)
6160eqcomd 2616 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑑 = 𝑐 ∧ (((𝐴𝑉𝐵𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩)) → ⟨𝐴, 𝑐, 𝐵⟩ = 𝑃)
6261fveq2d 6107 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑑 = 𝑐 ∧ (((𝐴𝑉𝐵𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩)) → (2nd ‘⟨𝐴, 𝑐, 𝐵⟩) = (2nd𝑃))
6358, 62eqtr3d 2646 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑑 = 𝑐 ∧ (((𝐴𝑉𝐵𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩)) → 𝐵 = (2nd𝑃))
6445, 54, 633eqtrd 2648 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑑 = 𝑐 ∧ (((𝐴𝑉𝐵𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩)) → (1st ‘(1st𝑄)) = (2nd𝑃))
6540, 41, 643jca 1235 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑑 = 𝑐 ∧ (((𝐴𝑉𝐵𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩)) → ((1st ‘(1st𝑃)) = (2nd𝑄) ∧ (2nd ‘(1st𝑃)) = (2nd ‘(1st𝑃)) ∧ (1st ‘(1st𝑄)) = (2nd𝑃)))
6665ex 449 . . . . . . . . . . . . . . . . . . . . 21 (𝑑 = 𝑐 → ((((𝐴𝑉𝐵𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩) → ((1st ‘(1st𝑃)) = (2nd𝑄) ∧ (2nd ‘(1st𝑃)) = (2nd ‘(1st𝑃)) ∧ (1st ‘(1st𝑄)) = (2nd𝑃))))
6722, 66syl6 34 . . . . . . . . . . . . . . . . . . . 20 ((𝑉 FriendGrph 𝐸𝐴𝐵) → ((⟨𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ⟨𝐵, 𝑑, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴)) → ((((𝐴𝑉𝐵𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩) → ((1st ‘(1st𝑃)) = (2nd𝑄) ∧ (2nd ‘(1st𝑃)) = (2nd ‘(1st𝑃)) ∧ (1st ‘(1st𝑄)) = (2nd𝑃)))))
6867expd 451 . . . . . . . . . . . . . . . . . . 19 ((𝑉 FriendGrph 𝐸𝐴𝐵) → (⟨𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) → (⟨𝐵, 𝑑, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴) → ((((𝐴𝑉𝐵𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩) → ((1st ‘(1st𝑃)) = (2nd𝑄) ∧ (2nd ‘(1st𝑃)) = (2nd ‘(1st𝑃)) ∧ (1st ‘(1st𝑄)) = (2nd𝑃))))))
6968com14 94 . . . . . . . . . . . . . . . . . 18 ((((𝐴𝑉𝐵𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩) → (⟨𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) → (⟨𝐵, 𝑑, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴) → ((𝑉 FriendGrph 𝐸𝐴𝐵) → ((1st ‘(1st𝑃)) = (2nd𝑄) ∧ (2nd ‘(1st𝑃)) = (2nd ‘(1st𝑃)) ∧ (1st ‘(1st𝑄)) = (2nd𝑃))))))
7069ex 449 . . . . . . . . . . . . . . . . 17 (((𝐴𝑉𝐵𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) → (𝑃 = ⟨𝐴, 𝑐, 𝐵⟩ → (⟨𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) → (⟨𝐵, 𝑑, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴) → ((𝑉 FriendGrph 𝐸𝐴𝐵) → ((1st ‘(1st𝑃)) = (2nd𝑄) ∧ (2nd ‘(1st𝑃)) = (2nd ‘(1st𝑃)) ∧ (1st ‘(1st𝑄)) = (2nd𝑃)))))))
7170ex 449 . . . . . . . . . . . . . . . 16 ((𝐴𝑉𝐵𝑉) → (𝑄 = ⟨𝐵, 𝑑, 𝐴⟩ → (𝑃 = ⟨𝐴, 𝑐, 𝐵⟩ → (⟨𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) → (⟨𝐵, 𝑑, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴) → ((𝑉 FriendGrph 𝐸𝐴𝐵) → ((1st ‘(1st𝑃)) = (2nd𝑄) ∧ (2nd ‘(1st𝑃)) = (2nd ‘(1st𝑃)) ∧ (1st ‘(1st𝑄)) = (2nd𝑃))))))))
72713ad2ant2 1076 . . . . . . . . . . . . . . 15 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)) → (𝑄 = ⟨𝐵, 𝑑, 𝐴⟩ → (𝑃 = ⟨𝐴, 𝑐, 𝐵⟩ → (⟨𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) → (⟨𝐵, 𝑑, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴) → ((𝑉 FriendGrph 𝐸𝐴𝐵) → ((1st ‘(1st𝑃)) = (2nd𝑄) ∧ (2nd ‘(1st𝑃)) = (2nd ‘(1st𝑃)) ∧ (1st ‘(1st𝑄)) = (2nd𝑃))))))))
7372ad2antrr 758 . . . . . . . . . . . . . 14 (((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)) ∧ 𝑐𝑉) ∧ 𝑑𝑉) → (𝑄 = ⟨𝐵, 𝑑, 𝐴⟩ → (𝑃 = ⟨𝐴, 𝑐, 𝐵⟩ → (⟨𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) → (⟨𝐵, 𝑑, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴) → ((𝑉 FriendGrph 𝐸𝐴𝐵) → ((1st ‘(1st𝑃)) = (2nd𝑄) ∧ (2nd ‘(1st𝑃)) = (2nd ‘(1st𝑃)) ∧ (1st ‘(1st𝑄)) = (2nd𝑃))))))))
7473imp31 447 . . . . . . . . . . . . 13 (((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)) ∧ 𝑐𝑉) ∧ 𝑑𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩) → (⟨𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) → (⟨𝐵, 𝑑, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴) → ((𝑉 FriendGrph 𝐸𝐴𝐵) → ((1st ‘(1st𝑃)) = (2nd𝑄) ∧ (2nd ‘(1st𝑃)) = (2nd ‘(1st𝑃)) ∧ (1st ‘(1st𝑄)) = (2nd𝑃))))))
7521, 74sylbid 229 . . . . . . . . . . . 12 (((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)) ∧ 𝑐𝑉) ∧ 𝑑𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩) → (∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑐 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2))) → (⟨𝐵, 𝑑, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴) → ((𝑉 FriendGrph 𝐸𝐴𝐵) → ((1st ‘(1st𝑃)) = (2nd𝑄) ∧ (2nd ‘(1st𝑃)) = (2nd ‘(1st𝑃)) ∧ (1st ‘(1st𝑄)) = (2nd𝑃))))))
7675expimpd 627 . . . . . . . . . . 11 ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)) ∧ 𝑐𝑉) ∧ 𝑑𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) → ((𝑃 = ⟨𝐴, 𝑐, 𝐵⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑐 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2)))) → (⟨𝐵, 𝑑, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴) → ((𝑉 FriendGrph 𝐸𝐴𝐵) → ((1st ‘(1st𝑃)) = (2nd𝑄) ∧ (2nd ‘(1st𝑃)) = (2nd ‘(1st𝑃)) ∧ (1st ‘(1st𝑄)) = (2nd𝑃))))))
7776com23 84 . . . . . . . . . 10 ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)) ∧ 𝑐𝑉) ∧ 𝑑𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) → (⟨𝐵, 𝑑, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴) → ((𝑃 = ⟨𝐴, 𝑐, 𝐵⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑐 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2)))) → ((𝑉 FriendGrph 𝐸𝐴𝐵) → ((1st ‘(1st𝑃)) = (2nd𝑄) ∧ (2nd ‘(1st𝑃)) = (2nd ‘(1st𝑃)) ∧ (1st ‘(1st𝑄)) = (2nd𝑃))))))
7815, 77sylbid 229 . . . . . . . . 9 ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)) ∧ 𝑐𝑉) ∧ 𝑑𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) → (∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐵 = (𝑝‘0) ∧ 𝑑 = (𝑝‘1) ∧ 𝐴 = (𝑝‘2))) → ((𝑃 = ⟨𝐴, 𝑐, 𝐵⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑐 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2)))) → ((𝑉 FriendGrph 𝐸𝐴𝐵) → ((1st ‘(1st𝑃)) = (2nd𝑄) ∧ (2nd ‘(1st𝑃)) = (2nd ‘(1st𝑃)) ∧ (1st ‘(1st𝑄)) = (2nd𝑃))))))
7978expimpd 627 . . . . . . . 8 (((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)) ∧ 𝑐𝑉) ∧ 𝑑𝑉) → ((𝑄 = ⟨𝐵, 𝑑, 𝐴⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐵 = (𝑝‘0) ∧ 𝑑 = (𝑝‘1) ∧ 𝐴 = (𝑝‘2)))) → ((𝑃 = ⟨𝐴, 𝑐, 𝐵⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑐 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2)))) → ((𝑉 FriendGrph 𝐸𝐴𝐵) → ((1st ‘(1st𝑃)) = (2nd𝑄) ∧ (2nd ‘(1st𝑃)) = (2nd ‘(1st𝑃)) ∧ (1st ‘(1st𝑄)) = (2nd𝑃))))))
8079rexlimdva 3013 . . . . . . 7 ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)) ∧ 𝑐𝑉) → (∃𝑑𝑉 (𝑄 = ⟨𝐵, 𝑑, 𝐴⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐵 = (𝑝‘0) ∧ 𝑑 = (𝑝‘1) ∧ 𝐴 = (𝑝‘2)))) → ((𝑃 = ⟨𝐴, 𝑐, 𝐵⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑐 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2)))) → ((𝑉 FriendGrph 𝐸𝐴𝐵) → ((1st ‘(1st𝑃)) = (2nd𝑄) ∧ (2nd ‘(1st𝑃)) = (2nd ‘(1st𝑃)) ∧ (1st ‘(1st𝑄)) = (2nd𝑃))))))
8180com23 84 . . . . . 6 ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)) ∧ 𝑐𝑉) → ((𝑃 = ⟨𝐴, 𝑐, 𝐵⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑐 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2)))) → (∃𝑑𝑉 (𝑄 = ⟨𝐵, 𝑑, 𝐴⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐵 = (𝑝‘0) ∧ 𝑑 = (𝑝‘1) ∧ 𝐴 = (𝑝‘2)))) → ((𝑉 FriendGrph 𝐸𝐴𝐵) → ((1st ‘(1st𝑃)) = (2nd𝑄) ∧ (2nd ‘(1st𝑃)) = (2nd ‘(1st𝑃)) ∧ (1st ‘(1st𝑄)) = (2nd𝑃))))))
8281rexlimdva 3013 . . . . 5 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)) → (∃𝑐𝑉 (𝑃 = ⟨𝐴, 𝑐, 𝐵⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑐 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2)))) → (∃𝑑𝑉 (𝑄 = ⟨𝐵, 𝑑, 𝐴⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐵 = (𝑝‘0) ∧ 𝑑 = (𝑝‘1) ∧ 𝐴 = (𝑝‘2)))) → ((𝑉 FriendGrph 𝐸𝐴𝐵) → ((1st ‘(1st𝑃)) = (2nd𝑄) ∧ (2nd ‘(1st𝑃)) = (2nd ‘(1st𝑃)) ∧ (1st ‘(1st𝑄)) = (2nd𝑃))))))
8382impd 446 . . . 4 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)) → ((∃𝑐𝑉 (𝑃 = ⟨𝐴, 𝑐, 𝐵⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑐 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2)))) ∧ ∃𝑑𝑉 (𝑄 = ⟨𝐵, 𝑑, 𝐴⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐵 = (𝑝‘0) ∧ 𝑑 = (𝑝‘1) ∧ 𝐴 = (𝑝‘2))))) → ((𝑉 FriendGrph 𝐸𝐴𝐵) → ((1st ‘(1st𝑃)) = (2nd𝑄) ∧ (2nd ‘(1st𝑃)) = (2nd ‘(1st𝑃)) ∧ (1st ‘(1st𝑄)) = (2nd𝑃)))))
849, 83sylbid 229 . . 3 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)) → ((𝑃 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ 𝑄 ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴)) → ((𝑉 FriendGrph 𝐸𝐴𝐵) → ((1st ‘(1st𝑃)) = (2nd𝑄) ∧ (2nd ‘(1st𝑃)) = (2nd ‘(1st𝑃)) ∧ (1st ‘(1st𝑄)) = (2nd𝑃)))))
852, 84mpcom 37 . 2 ((𝑃 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ 𝑄 ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴)) → ((𝑉 FriendGrph 𝐸𝐴𝐵) → ((1st ‘(1st𝑃)) = (2nd𝑄) ∧ (2nd ‘(1st𝑃)) = (2nd ‘(1st𝑃)) ∧ (1st ‘(1st𝑄)) = (2nd𝑃))))
8685com12 32 1 ((𝑉 FriendGrph 𝐸𝐴𝐵) → ((𝑃 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ 𝑄 ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴)) → ((1st ‘(1st𝑃)) = (2nd𝑄) ∧ (2nd ‘(1st𝑃)) = (2nd ‘(1st𝑃)) ∧ (1st ‘(1st𝑄)) = (2nd𝑃))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  Vcvv 3173  ⟨cotp 4133   class class class wbr 4583   × cxp 5036  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  0cc0 9815  1c1 9816  2c2 10947  #chash 12979   Walks cwalk 26026   2WalksOnOt c2wlkonot 26382   FriendGrph cfrgra 26515 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-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  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-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-ot 4134  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-usgra 25862  df-wlk 26036  df-wlkon 26042  df-2wlkonot 26385  df-frgra 26516 This theorem is referenced by: (None)
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