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Theorem el2wlkonotot0 26399
 Description: A walk of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.)
Assertion
Ref Expression
el2wlkonotot0 (((𝑉𝑋𝐸𝑌) ∧ (𝑅𝑉𝑆𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝑅(𝑉 2WalksOnOt 𝐸)𝑆) ↔ (𝐴 = 𝑅𝐶 = 𝑆 ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
Distinct variable groups:   𝐴,𝑓,𝑝   𝐵,𝑓,𝑝   𝐶,𝑓,𝑝   𝑓,𝐸,𝑝   𝑓,𝑉,𝑝   𝑅,𝑓,𝑝   𝑆,𝑓,𝑝   𝑓,𝑋,𝑝   𝑓,𝑌,𝑝

Proof of Theorem el2wlkonotot0
Dummy variable 𝑏 is distinct from all other variables.
StepHypRef Expression
1 el2wlkonot 26396 . 2 (((𝑉𝑋𝐸𝑌) ∧ (𝑅𝑉𝑆𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝑅(𝑉 2WalksOnOt 𝐸)𝑆) ↔ ∃𝑏𝑉 (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2))))))
2 19.42vv 1907 . . . . . 6 (∃𝑓𝑝(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))) ↔ (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))))
32bicomi 213 . . . . 5 ((⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))) ↔ ∃𝑓𝑝(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))))
43rexbii 3023 . . . 4 (∃𝑏𝑉 (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))) ↔ ∃𝑏𝑉𝑓𝑝(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))))
5 rexcom4 3198 . . . 4 (∃𝑏𝑉𝑓𝑝(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))) ↔ ∃𝑓𝑏𝑉𝑝(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))))
6 rexcom4 3198 . . . . 5 (∃𝑏𝑉𝑝(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))) ↔ ∃𝑝𝑏𝑉 (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))))
76exbii 1764 . . . 4 (∃𝑓𝑏𝑉𝑝(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))) ↔ ∃𝑓𝑝𝑏𝑉 (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))))
84, 5, 73bitri 285 . . 3 (∃𝑏𝑉 (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))) ↔ ∃𝑓𝑝𝑏𝑉 (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))))
98a1i 11 . 2 (((𝑉𝑋𝐸𝑌) ∧ (𝑅𝑉𝑆𝑉)) → (∃𝑏𝑉 (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))) ↔ ∃𝑓𝑝𝑏𝑉 (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2))))))
10 eqcom 2617 . . . . . . . . . . 11 (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ↔ ⟨𝑅, 𝑏, 𝑆⟩ = ⟨𝐴, 𝐵, 𝐶⟩)
11 df-ot 4134 . . . . . . . . . . . 12 𝑅, 𝑏, 𝑆⟩ = ⟨⟨𝑅, 𝑏⟩, 𝑆
12 df-ot 4134 . . . . . . . . . . . 12 𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶
1311, 12eqeq12i 2624 . . . . . . . . . . 11 (⟨𝑅, 𝑏, 𝑆⟩ = ⟨𝐴, 𝐵, 𝐶⟩ ↔ ⟨⟨𝑅, 𝑏⟩, 𝑆⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩)
1410, 13bitri 263 . . . . . . . . . 10 (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ↔ ⟨⟨𝑅, 𝑏⟩, 𝑆⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩)
1514a1i 11 . . . . . . . . 9 ((((𝑉𝑋𝐸𝑌) ∧ (𝑅𝑉𝑆𝑉)) ∧ 𝑏𝑉) → (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ↔ ⟨⟨𝑅, 𝑏⟩, 𝑆⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩))
16 opex 4859 . . . . . . . . . . . . . 14 𝑅, 𝑏⟩ ∈ V
1716a1i 11 . . . . . . . . . . . . 13 ((𝑅𝑉𝑆𝑉) → ⟨𝑅, 𝑏⟩ ∈ V)
18 simpr 476 . . . . . . . . . . . . 13 ((𝑅𝑉𝑆𝑉) → 𝑆𝑉)
1917, 18jca 553 . . . . . . . . . . . 12 ((𝑅𝑉𝑆𝑉) → (⟨𝑅, 𝑏⟩ ∈ V ∧ 𝑆𝑉))
2019adantl 481 . . . . . . . . . . 11 (((𝑉𝑋𝐸𝑌) ∧ (𝑅𝑉𝑆𝑉)) → (⟨𝑅, 𝑏⟩ ∈ V ∧ 𝑆𝑉))
2120adantr 480 . . . . . . . . . 10 ((((𝑉𝑋𝐸𝑌) ∧ (𝑅𝑉𝑆𝑉)) ∧ 𝑏𝑉) → (⟨𝑅, 𝑏⟩ ∈ V ∧ 𝑆𝑉))
22 opthg 4872 . . . . . . . . . 10 ((⟨𝑅, 𝑏⟩ ∈ V ∧ 𝑆𝑉) → (⟨⟨𝑅, 𝑏⟩, 𝑆⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ↔ (⟨𝑅, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ 𝑆 = 𝐶)))
2321, 22syl 17 . . . . . . . . 9 ((((𝑉𝑋𝐸𝑌) ∧ (𝑅𝑉𝑆𝑉)) ∧ 𝑏𝑉) → (⟨⟨𝑅, 𝑏⟩, 𝑆⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ↔ (⟨𝑅, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ 𝑆 = 𝐶)))
24 simpl 472 . . . . . . . . . . . 12 ((𝑅𝑉𝑆𝑉) → 𝑅𝑉)
2524adantl 481 . . . . . . . . . . 11 (((𝑉𝑋𝐸𝑌) ∧ (𝑅𝑉𝑆𝑉)) → 𝑅𝑉)
26 opthg 4872 . . . . . . . . . . 11 ((𝑅𝑉𝑏𝑉) → (⟨𝑅, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑅 = 𝐴𝑏 = 𝐵)))
2725, 26sylan 487 . . . . . . . . . 10 ((((𝑉𝑋𝐸𝑌) ∧ (𝑅𝑉𝑆𝑉)) ∧ 𝑏𝑉) → (⟨𝑅, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑅 = 𝐴𝑏 = 𝐵)))
2827anbi1d 737 . . . . . . . . 9 ((((𝑉𝑋𝐸𝑌) ∧ (𝑅𝑉𝑆𝑉)) ∧ 𝑏𝑉) → ((⟨𝑅, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ 𝑆 = 𝐶) ↔ ((𝑅 = 𝐴𝑏 = 𝐵) ∧ 𝑆 = 𝐶)))
2915, 23, 283bitrd 293 . . . . . . . 8 ((((𝑉𝑋𝐸𝑌) ∧ (𝑅𝑉𝑆𝑉)) ∧ 𝑏𝑉) → (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ↔ ((𝑅 = 𝐴𝑏 = 𝐵) ∧ 𝑆 = 𝐶)))
30 eqcom 2617 . . . . . . . . . . . . . 14 (𝑅 = 𝐴𝐴 = 𝑅)
3130biimpi 205 . . . . . . . . . . . . 13 (𝑅 = 𝐴𝐴 = 𝑅)
3231adantr 480 . . . . . . . . . . . 12 ((𝑅 = 𝐴𝑏 = 𝐵) → 𝐴 = 𝑅)
33 eqcom 2617 . . . . . . . . . . . . 13 (𝑆 = 𝐶𝐶 = 𝑆)
3433biimpi 205 . . . . . . . . . . . 12 (𝑆 = 𝐶𝐶 = 𝑆)
3532, 34anim12i 588 . . . . . . . . . . 11 (((𝑅 = 𝐴𝑏 = 𝐵) ∧ 𝑆 = 𝐶) → (𝐴 = 𝑅𝐶 = 𝑆))
3635adantr 480 . . . . . . . . . 10 ((((𝑅 = 𝐴𝑏 = 𝐵) ∧ 𝑆 = 𝐶) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))) → (𝐴 = 𝑅𝐶 = 𝑆))
37 simpr1 1060 . . . . . . . . . . 11 ((((𝑅 = 𝐴𝑏 = 𝐵) ∧ 𝑆 = 𝐶) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))) → 𝑓(𝑉 Walks 𝐸)𝑝)
38 simpr2 1061 . . . . . . . . . . 11 ((((𝑅 = 𝐴𝑏 = 𝐵) ∧ 𝑆 = 𝐶) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))) → (#‘𝑓) = 2)
39 eqtr2 2630 . . . . . . . . . . . . . . . . 17 ((𝑅 = 𝐴𝑅 = (𝑝‘0)) → 𝐴 = (𝑝‘0))
4039ex 449 . . . . . . . . . . . . . . . 16 (𝑅 = 𝐴 → (𝑅 = (𝑝‘0) → 𝐴 = (𝑝‘0)))
4140ad2antrr 758 . . . . . . . . . . . . . . 15 (((𝑅 = 𝐴𝑏 = 𝐵) ∧ 𝑆 = 𝐶) → (𝑅 = (𝑝‘0) → 𝐴 = (𝑝‘0)))
42 eqtr2 2630 . . . . . . . . . . . . . . . . . 18 ((𝑏 = 𝐵𝑏 = (𝑝‘1)) → 𝐵 = (𝑝‘1))
4342ex 449 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝐵 → (𝑏 = (𝑝‘1) → 𝐵 = (𝑝‘1)))
4443adantl 481 . . . . . . . . . . . . . . . 16 ((𝑅 = 𝐴𝑏 = 𝐵) → (𝑏 = (𝑝‘1) → 𝐵 = (𝑝‘1)))
4544adantr 480 . . . . . . . . . . . . . . 15 (((𝑅 = 𝐴𝑏 = 𝐵) ∧ 𝑆 = 𝐶) → (𝑏 = (𝑝‘1) → 𝐵 = (𝑝‘1)))
46 eqtr2 2630 . . . . . . . . . . . . . . . . 17 ((𝑆 = 𝐶𝑆 = (𝑝‘2)) → 𝐶 = (𝑝‘2))
4746ex 449 . . . . . . . . . . . . . . . 16 (𝑆 = 𝐶 → (𝑆 = (𝑝‘2) → 𝐶 = (𝑝‘2)))
4847adantl 481 . . . . . . . . . . . . . . 15 (((𝑅 = 𝐴𝑏 = 𝐵) ∧ 𝑆 = 𝐶) → (𝑆 = (𝑝‘2) → 𝐶 = (𝑝‘2)))
4941, 45, 483anim123d 1398 . . . . . . . . . . . . . 14 (((𝑅 = 𝐴𝑏 = 𝐵) ∧ 𝑆 = 𝐶) → ((𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)) → (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))
5049com12 32 . . . . . . . . . . . . 13 ((𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)) → (((𝑅 = 𝐴𝑏 = 𝐵) ∧ 𝑆 = 𝐶) → (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))
51503ad2ant3 1077 . . . . . . . . . . . 12 ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2))) → (((𝑅 = 𝐴𝑏 = 𝐵) ∧ 𝑆 = 𝐶) → (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))
5251impcom 445 . . . . . . . . . . 11 ((((𝑅 = 𝐴𝑏 = 𝐵) ∧ 𝑆 = 𝐶) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))) → (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))
5337, 38, 523jca 1235 . . . . . . . . . 10 ((((𝑅 = 𝐴𝑏 = 𝐵) ∧ 𝑆 = 𝐶) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))) → (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))
5436, 53jca 553 . . . . . . . . 9 ((((𝑅 = 𝐴𝑏 = 𝐵) ∧ 𝑆 = 𝐶) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))) → ((𝐴 = 𝑅𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
5554ex 449 . . . . . . . 8 (((𝑅 = 𝐴𝑏 = 𝐵) ∧ 𝑆 = 𝐶) → ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2))) → ((𝐴 = 𝑅𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
5629, 55syl6bi 242 . . . . . . 7 ((((𝑉𝑋𝐸𝑌) ∧ (𝑅𝑉𝑆𝑉)) ∧ 𝑏𝑉) → (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ → ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2))) → ((𝐴 = 𝑅𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))))
5756impd 446 . . . . . 6 ((((𝑉𝑋𝐸𝑌) ∧ (𝑅𝑉𝑆𝑉)) ∧ 𝑏𝑉) → ((⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))) → ((𝐴 = 𝑅𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
5857rexlimdva 3013 . . . . 5 (((𝑉𝑋𝐸𝑌) ∧ (𝑅𝑉𝑆𝑉)) → (∃𝑏𝑉 (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))) → ((𝐴 = 𝑅𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
59 el2wlkonotlem 26389 . . . . . . . . . . . . . . . . 17 ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) → (𝑝‘1) ∈ 𝑉)
6059adantr 480 . . . . . . . . . . . . . . . 16 (((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) ∧ 𝐵 = (𝑝‘1)) → (𝑝‘1) ∈ 𝑉)
61 eleq1 2676 . . . . . . . . . . . . . . . . 17 (𝐵 = (𝑝‘1) → (𝐵𝑉 ↔ (𝑝‘1) ∈ 𝑉))
6261adantl 481 . . . . . . . . . . . . . . . 16 (((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) ∧ 𝐵 = (𝑝‘1)) → (𝐵𝑉 ↔ (𝑝‘1) ∈ 𝑉))
6360, 62mpbird 246 . . . . . . . . . . . . . . 15 (((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) ∧ 𝐵 = (𝑝‘1)) → 𝐵𝑉)
6463a1d 25 . . . . . . . . . . . . . 14 (((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) ∧ 𝐵 = (𝑝‘1)) → ((𝐴 = 𝑅𝐶 = 𝑆) → 𝐵𝑉))
6564ex 449 . . . . . . . . . . . . 13 ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) → (𝐵 = (𝑝‘1) → ((𝐴 = 𝑅𝐶 = 𝑆) → 𝐵𝑉)))
6665ex 449 . . . . . . . . . . . 12 (𝑓(𝑉 Walks 𝐸)𝑝 → ((#‘𝑓) = 2 → (𝐵 = (𝑝‘1) → ((𝐴 = 𝑅𝐶 = 𝑆) → 𝐵𝑉))))
6766com13 86 . . . . . . . . . . 11 (𝐵 = (𝑝‘1) → ((#‘𝑓) = 2 → (𝑓(𝑉 Walks 𝐸)𝑝 → ((𝐴 = 𝑅𝐶 = 𝑆) → 𝐵𝑉))))
68673ad2ant2 1076 . . . . . . . . . 10 ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → ((#‘𝑓) = 2 → (𝑓(𝑉 Walks 𝐸)𝑝 → ((𝐴 = 𝑅𝐶 = 𝑆) → 𝐵𝑉))))
6968com13 86 . . . . . . . . 9 (𝑓(𝑉 Walks 𝐸)𝑝 → ((#‘𝑓) = 2 → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → ((𝐴 = 𝑅𝐶 = 𝑆) → 𝐵𝑉))))
70693imp 1249 . . . . . . . 8 ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → ((𝐴 = 𝑅𝐶 = 𝑆) → 𝐵𝑉))
7170impcom 445 . . . . . . 7 (((𝐴 = 𝑅𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → 𝐵𝑉)
72 simpl 472 . . . . . . . . . . 11 ((𝐴 = 𝑅𝐶 = 𝑆) → 𝐴 = 𝑅)
7372ad2antrr 758 . . . . . . . . . 10 ((((𝐴 = 𝑅𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ∧ 𝑏 = 𝐵) → 𝐴 = 𝑅)
74 eqcom 2617 . . . . . . . . . . . 12 (𝑏 = 𝐵𝐵 = 𝑏)
7574biimpi 205 . . . . . . . . . . 11 (𝑏 = 𝐵𝐵 = 𝑏)
7675adantl 481 . . . . . . . . . 10 ((((𝐴 = 𝑅𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ∧ 𝑏 = 𝐵) → 𝐵 = 𝑏)
77 simpr 476 . . . . . . . . . . 11 ((𝐴 = 𝑅𝐶 = 𝑆) → 𝐶 = 𝑆)
7877ad2antrr 758 . . . . . . . . . 10 ((((𝐴 = 𝑅𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ∧ 𝑏 = 𝐵) → 𝐶 = 𝑆)
7973, 76, 78oteq123d 4355 . . . . . . . . 9 ((((𝐴 = 𝑅𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ∧ 𝑏 = 𝐵) → ⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩)
80 simpr1 1060 . . . . . . . . . . 11 (((𝐴 = 𝑅𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → 𝑓(𝑉 Walks 𝐸)𝑝)
8180adantr 480 . . . . . . . . . 10 ((((𝐴 = 𝑅𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ∧ 𝑏 = 𝐵) → 𝑓(𝑉 Walks 𝐸)𝑝)
82 simplr2 1097 . . . . . . . . . 10 ((((𝐴 = 𝑅𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ∧ 𝑏 = 𝐵) → (#‘𝑓) = 2)
83 eqtr2 2630 . . . . . . . . . . . . . . . . . . 19 ((𝐴 = 𝑅𝐴 = (𝑝‘0)) → 𝑅 = (𝑝‘0))
8483ex 449 . . . . . . . . . . . . . . . . . 18 (𝐴 = 𝑅 → (𝐴 = (𝑝‘0) → 𝑅 = (𝑝‘0)))
8584adantr 480 . . . . . . . . . . . . . . . . 17 ((𝐴 = 𝑅𝐶 = 𝑆) → (𝐴 = (𝑝‘0) → 𝑅 = (𝑝‘0)))
8685adantl 481 . . . . . . . . . . . . . . . 16 ((𝑏 = 𝐵 ∧ (𝐴 = 𝑅𝐶 = 𝑆)) → (𝐴 = (𝑝‘0) → 𝑅 = (𝑝‘0)))
87 eqtr2 2630 . . . . . . . . . . . . . . . . . . 19 ((𝐵 = 𝑏𝐵 = (𝑝‘1)) → 𝑏 = (𝑝‘1))
8887ex 449 . . . . . . . . . . . . . . . . . 18 (𝐵 = 𝑏 → (𝐵 = (𝑝‘1) → 𝑏 = (𝑝‘1)))
8988eqcoms 2618 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝐵 → (𝐵 = (𝑝‘1) → 𝑏 = (𝑝‘1)))
9089adantr 480 . . . . . . . . . . . . . . . 16 ((𝑏 = 𝐵 ∧ (𝐴 = 𝑅𝐶 = 𝑆)) → (𝐵 = (𝑝‘1) → 𝑏 = (𝑝‘1)))
91 eqtr2 2630 . . . . . . . . . . . . . . . . . . 19 ((𝐶 = 𝑆𝐶 = (𝑝‘2)) → 𝑆 = (𝑝‘2))
9291ex 449 . . . . . . . . . . . . . . . . . 18 (𝐶 = 𝑆 → (𝐶 = (𝑝‘2) → 𝑆 = (𝑝‘2)))
9392adantl 481 . . . . . . . . . . . . . . . . 17 ((𝐴 = 𝑅𝐶 = 𝑆) → (𝐶 = (𝑝‘2) → 𝑆 = (𝑝‘2)))
9493adantl 481 . . . . . . . . . . . . . . . 16 ((𝑏 = 𝐵 ∧ (𝐴 = 𝑅𝐶 = 𝑆)) → (𝐶 = (𝑝‘2) → 𝑆 = (𝑝‘2)))
9586, 90, 943anim123d 1398 . . . . . . . . . . . . . . 15 ((𝑏 = 𝐵 ∧ (𝐴 = 𝑅𝐶 = 𝑆)) → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2))))
9695ex 449 . . . . . . . . . . . . . 14 (𝑏 = 𝐵 → ((𝐴 = 𝑅𝐶 = 𝑆) → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))))
9796com13 86 . . . . . . . . . . . . 13 ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → ((𝐴 = 𝑅𝐶 = 𝑆) → (𝑏 = 𝐵 → (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))))
98973ad2ant3 1077 . . . . . . . . . . . 12 ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → ((𝐴 = 𝑅𝐶 = 𝑆) → (𝑏 = 𝐵 → (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))))
9998impcom 445 . . . . . . . . . . 11 (((𝐴 = 𝑅𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (𝑏 = 𝐵 → (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2))))
10099imp 444 . . . . . . . . . 10 ((((𝐴 = 𝑅𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ∧ 𝑏 = 𝐵) → (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))
10181, 82, 1003jca 1235 . . . . . . . . 9 ((((𝐴 = 𝑅𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ∧ 𝑏 = 𝐵) → (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2))))
10279, 101jca 553 . . . . . . . 8 ((((𝐴 = 𝑅𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ∧ 𝑏 = 𝐵) → (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))))
103102a1d 25 . . . . . . 7 ((((𝐴 = 𝑅𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ∧ 𝑏 = 𝐵) → (((𝑉𝑋𝐸𝑌) ∧ (𝑅𝑉𝑆𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2))))))
10471, 103rspcimedv 3284 . . . . . 6 (((𝐴 = 𝑅𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (((𝑉𝑋𝐸𝑌) ∧ (𝑅𝑉𝑆𝑉)) → ∃𝑏𝑉 (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2))))))
105104com12 32 . . . . 5 (((𝑉𝑋𝐸𝑌) ∧ (𝑅𝑉𝑆𝑉)) → (((𝐴 = 𝑅𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → ∃𝑏𝑉 (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2))))))
10658, 105impbid 201 . . . 4 (((𝑉𝑋𝐸𝑌) ∧ (𝑅𝑉𝑆𝑉)) → (∃𝑏𝑉 (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))) ↔ ((𝐴 = 𝑅𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
1071062exbidv 1839 . . 3 (((𝑉𝑋𝐸𝑌) ∧ (𝑅𝑉𝑆𝑉)) → (∃𝑓𝑝𝑏𝑉 (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))) ↔ ∃𝑓𝑝((𝐴 = 𝑅𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
108 df-3an 1033 . . . 4 ((𝐴 = 𝑅𝐶 = 𝑆 ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ↔ ((𝐴 = 𝑅𝐶 = 𝑆) ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
109 19.42vv 1907 . . . . 5 (∃𝑓𝑝((𝐴 = 𝑅𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ↔ ((𝐴 = 𝑅𝐶 = 𝑆) ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
110109bicomi 213 . . . 4 (((𝐴 = 𝑅𝐶 = 𝑆) ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ↔ ∃𝑓𝑝((𝐴 = 𝑅𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
111108, 110bitri 263 . . 3 ((𝐴 = 𝑅𝐶 = 𝑆 ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ↔ ∃𝑓𝑝((𝐴 = 𝑅𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
112107, 111syl6bbr 277 . 2 (((𝑉𝑋𝐸𝑌) ∧ (𝑅𝑉𝑆𝑉)) → (∃𝑓𝑝𝑏𝑉 (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))) ↔ (𝐴 = 𝑅𝐶 = 𝑆 ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
1131, 9, 1123bitrd 293 1 (((𝑉𝑋𝐸𝑌) ∧ (𝑅𝑉𝑆𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝑅(𝑉 2WalksOnOt 𝐸)𝑆) ↔ (𝐴 = 𝑅𝐶 = 𝑆 ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173  ⟨cop 4131  ⟨cotp 4133   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816  2c2 10947  #chash 12979   Walks cwalk 26026   2WalksOnOt c2wlkonot 26382 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-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-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  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-wlk 26036  df-wlkon 26042  df-2wlkonot 26385 This theorem is referenced by:  el2wlkonotot  26400  el2wlkonotot1  26401  el2wlksotot  26409
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