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Theorem elwwlks2ons3 26711
 Description: For each walk of length 2 between two vertices, there is a third vertex in the middle of the walk. (Contributed by Alexander van der Vekens, 15-Feb-2018.) (Revised by AV, 12-May-2021.)
Hypothesis
Ref Expression
elwwlks2ons3.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
elwwlks2ons3 ((𝐺𝑈𝐴𝑉𝐶𝑉) → (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ∃𝑏𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))))
Distinct variable groups:   𝐴,𝑏   𝐶,𝑏   𝐺,𝑏   𝑈,𝑏   𝑉,𝑏   𝑊,𝑏

Proof of Theorem elwwlks2ons3
StepHypRef Expression
1 simpr 477 . . . . 5 (((𝐺𝑈𝐴𝑉𝐶𝑉) ∧ 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))
2 elwwlks2ons3.v . . . . . . . . 9 𝑉 = (Vtx‘𝐺)
32wwlknon 26605 . . . . . . . 8 ((𝐴𝑉𝐶𝑉) → (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ (𝑊 ∈ (2 WWalksN 𝐺) ∧ (𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶)))
433adant1 1077 . . . . . . 7 ((𝐺𝑈𝐴𝑉𝐶𝑉) → (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ (𝑊 ∈ (2 WWalksN 𝐺) ∧ (𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶)))
5 wwlknbp2 26615 . . . . . . . . . 10 (𝑊 ∈ (2 WWalksN 𝐺) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (2 + 1)))
6 2p1e3 11096 . . . . . . . . . . . 12 (2 + 1) = 3
76eqeq2i 2638 . . . . . . . . . . 11 ((#‘𝑊) = (2 + 1) ↔ (#‘𝑊) = 3)
8 1ex 9980 . . . . . . . . . . . . . . . . 17 1 ∈ V
98tpid2 4279 . . . . . . . . . . . . . . . 16 1 ∈ {0, 1, 2}
10 oveq2 6613 . . . . . . . . . . . . . . . . 17 ((#‘𝑊) = 3 → (0..^(#‘𝑊)) = (0..^3))
11 fzo0to3tp 12492 . . . . . . . . . . . . . . . . 17 (0..^3) = {0, 1, 2}
1210, 11syl6eq 2676 . . . . . . . . . . . . . . . 16 ((#‘𝑊) = 3 → (0..^(#‘𝑊)) = {0, 1, 2})
139, 12syl5eleqr 2711 . . . . . . . . . . . . . . 15 ((#‘𝑊) = 3 → 1 ∈ (0..^(#‘𝑊)))
14 wrdsymbcl 13252 . . . . . . . . . . . . . . 15 ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 1 ∈ (0..^(#‘𝑊))) → (𝑊‘1) ∈ (Vtx‘𝐺))
1513, 14sylan2 491 . . . . . . . . . . . . . 14 ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 3) → (𝑊‘1) ∈ (Vtx‘𝐺))
16153ad2ant1 1080 . . . . . . . . . . . . 13 (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐺𝑈𝐴𝑉𝐶𝑉)) → (𝑊‘1) ∈ (Vtx‘𝐺))
17 simpr 477 . . . . . . . . . . . . . . . . 17 ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 3) → (#‘𝑊) = 3)
18173ad2ant1 1080 . . . . . . . . . . . . . . . 16 (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐺𝑈𝐴𝑉𝐶𝑉)) → (#‘𝑊) = 3)
1918adantr 481 . . . . . . . . . . . . . . 15 ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐺𝑈𝐴𝑉𝐶𝑉)) ∧ (𝑊‘1) ∈ (Vtx‘𝐺)) → (#‘𝑊) = 3)
20 simpl 473 . . . . . . . . . . . . . . . . . 18 (((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) → (𝑊‘0) = 𝐴)
21 eqidd 2627 . . . . . . . . . . . . . . . . . 18 (((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) → (𝑊‘1) = (𝑊‘1))
22 simpr 477 . . . . . . . . . . . . . . . . . 18 (((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) → (𝑊‘2) = 𝐶)
2320, 21, 223jca 1240 . . . . . . . . . . . . . . . . 17 (((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) → ((𝑊‘0) = 𝐴 ∧ (𝑊‘1) = (𝑊‘1) ∧ (𝑊‘2) = 𝐶))
24233ad2ant2 1081 . . . . . . . . . . . . . . . 16 (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐺𝑈𝐴𝑉𝐶𝑉)) → ((𝑊‘0) = 𝐴 ∧ (𝑊‘1) = (𝑊‘1) ∧ (𝑊‘2) = 𝐶))
2524adantr 481 . . . . . . . . . . . . . . 15 ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐺𝑈𝐴𝑉𝐶𝑉)) ∧ (𝑊‘1) ∈ (Vtx‘𝐺)) → ((𝑊‘0) = 𝐴 ∧ (𝑊‘1) = (𝑊‘1) ∧ (𝑊‘2) = 𝐶))
262eqcomi 2635 . . . . . . . . . . . . . . . . . . . . . 22 (Vtx‘𝐺) = 𝑉
2726wrdeqi 13262 . . . . . . . . . . . . . . . . . . . . 21 Word (Vtx‘𝐺) = Word 𝑉
2827eleq2i 2696 . . . . . . . . . . . . . . . . . . . 20 (𝑊 ∈ Word (Vtx‘𝐺) ↔ 𝑊 ∈ Word 𝑉)
2928biimpi 206 . . . . . . . . . . . . . . . . . . 19 (𝑊 ∈ Word (Vtx‘𝐺) → 𝑊 ∈ Word 𝑉)
3029adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 3) → 𝑊 ∈ Word 𝑉)
31303ad2ant1 1080 . . . . . . . . . . . . . . . . 17 (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐺𝑈𝐴𝑉𝐶𝑉)) → 𝑊 ∈ Word 𝑉)
3231adantr 481 . . . . . . . . . . . . . . . 16 ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐺𝑈𝐴𝑉𝐶𝑉)) ∧ (𝑊‘1) ∈ (Vtx‘𝐺)) → 𝑊 ∈ Word 𝑉)
33 simpl32 1141 . . . . . . . . . . . . . . . 16 ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐺𝑈𝐴𝑉𝐶𝑉)) ∧ (𝑊‘1) ∈ (Vtx‘𝐺)) → 𝐴𝑉)
3426eleq2i 2696 . . . . . . . . . . . . . . . . . 18 ((𝑊‘1) ∈ (Vtx‘𝐺) ↔ (𝑊‘1) ∈ 𝑉)
3534biimpi 206 . . . . . . . . . . . . . . . . 17 ((𝑊‘1) ∈ (Vtx‘𝐺) → (𝑊‘1) ∈ 𝑉)
3635adantl 482 . . . . . . . . . . . . . . . 16 ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐺𝑈𝐴𝑉𝐶𝑉)) ∧ (𝑊‘1) ∈ (Vtx‘𝐺)) → (𝑊‘1) ∈ 𝑉)
37 simpl33 1142 . . . . . . . . . . . . . . . 16 ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐺𝑈𝐴𝑉𝐶𝑉)) ∧ (𝑊‘1) ∈ (Vtx‘𝐺)) → 𝐶𝑉)
38 eqwrds3 13633 . . . . . . . . . . . . . . . 16 ((𝑊 ∈ Word 𝑉 ∧ (𝐴𝑉 ∧ (𝑊‘1) ∈ 𝑉𝐶𝑉)) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ↔ ((#‘𝑊) = 3 ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘1) = (𝑊‘1) ∧ (𝑊‘2) = 𝐶))))
3932, 33, 36, 37, 38syl13anc 1325 . . . . . . . . . . . . . . 15 ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐺𝑈𝐴𝑉𝐶𝑉)) ∧ (𝑊‘1) ∈ (Vtx‘𝐺)) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ↔ ((#‘𝑊) = 3 ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘1) = (𝑊‘1) ∧ (𝑊‘2) = 𝐶))))
4019, 25, 39mpbir2and 956 . . . . . . . . . . . . . 14 ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐺𝑈𝐴𝑉𝐶𝑉)) ∧ (𝑊‘1) ∈ (Vtx‘𝐺)) → 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩)
4140, 36jca 554 . . . . . . . . . . . . 13 ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐺𝑈𝐴𝑉𝐶𝑉)) ∧ (𝑊‘1) ∈ (Vtx‘𝐺)) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ (𝑊‘1) ∈ 𝑉))
4216, 41mpdan 701 . . . . . . . . . . . 12 (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐺𝑈𝐴𝑉𝐶𝑉)) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ (𝑊‘1) ∈ 𝑉))
43423exp 1261 . . . . . . . . . . 11 ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 3) → (((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) → ((𝐺𝑈𝐴𝑉𝐶𝑉) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ (𝑊‘1) ∈ 𝑉))))
447, 43sylan2b 492 . . . . . . . . . 10 ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (2 + 1)) → (((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) → ((𝐺𝑈𝐴𝑉𝐶𝑉) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ (𝑊‘1) ∈ 𝑉))))
455, 44syl 17 . . . . . . . . 9 (𝑊 ∈ (2 WWalksN 𝐺) → (((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) → ((𝐺𝑈𝐴𝑉𝐶𝑉) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ (𝑊‘1) ∈ 𝑉))))
46453impib 1259 . . . . . . . 8 ((𝑊 ∈ (2 WWalksN 𝐺) ∧ (𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) → ((𝐺𝑈𝐴𝑉𝐶𝑉) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ (𝑊‘1) ∈ 𝑉)))
4746com12 32 . . . . . . 7 ((𝐺𝑈𝐴𝑉𝐶𝑉) → ((𝑊 ∈ (2 WWalksN 𝐺) ∧ (𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ (𝑊‘1) ∈ 𝑉)))
484, 47sylbid 230 . . . . . 6 ((𝐺𝑈𝐴𝑉𝐶𝑉) → (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ (𝑊‘1) ∈ 𝑉)))
4948imp 445 . . . . 5 (((𝐺𝑈𝐴𝑉𝐶𝑉) ∧ 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ (𝑊‘1) ∈ 𝑉))
50 anass 680 . . . . 5 (((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩) ∧ (𝑊‘1) ∈ 𝑉) ↔ (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ (𝑊‘1) ∈ 𝑉)))
511, 49, 50sylanbrc 697 . . . 4 (((𝐺𝑈𝐴𝑉𝐶𝑉) ∧ 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → ((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩) ∧ (𝑊‘1) ∈ 𝑉))
52 simpr 477 . . . . 5 (((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩) ∧ (𝑊‘1) ∈ 𝑉) → (𝑊‘1) ∈ 𝑉)
53 eqidd 2627 . . . . . . . 8 (𝑏 = (𝑊‘1) → 𝐴 = 𝐴)
54 id 22 . . . . . . . 8 (𝑏 = (𝑊‘1) → 𝑏 = (𝑊‘1))
55 eqidd 2627 . . . . . . . 8 (𝑏 = (𝑊‘1) → 𝐶 = 𝐶)
5653, 54, 55s3eqd 13541 . . . . . . 7 (𝑏 = (𝑊‘1) → ⟨“𝐴𝑏𝐶”⟩ = ⟨“𝐴(𝑊‘1)𝐶”⟩)
57 eqeq2 2637 . . . . . . . 8 (⟨“𝐴𝑏𝐶”⟩ = ⟨“𝐴(𝑊‘1)𝐶”⟩ → (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ↔ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩))
58 eleq1 2692 . . . . . . . 8 (⟨“𝐴𝑏𝐶”⟩ = ⟨“𝐴(𝑊‘1)𝐶”⟩ → (⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ⟨“𝐴(𝑊‘1)𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))
5957, 58anbi12d 746 . . . . . . 7 (⟨“𝐴𝑏𝐶”⟩ = ⟨“𝐴(𝑊‘1)𝐶”⟩ → ((𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) ↔ (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ ⟨“𝐴(𝑊‘1)𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))))
6056, 59syl 17 . . . . . 6 (𝑏 = (𝑊‘1) → ((𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) ↔ (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ ⟨“𝐴(𝑊‘1)𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))))
6160adantl 482 . . . . 5 ((((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩) ∧ (𝑊‘1) ∈ 𝑉) ∧ 𝑏 = (𝑊‘1)) → ((𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) ↔ (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ ⟨“𝐴(𝑊‘1)𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))))
62 simpr 477 . . . . . . 7 ((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩) → 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩)
63 eleq1 2692 . . . . . . . 8 (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ → (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ⟨“𝐴(𝑊‘1)𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))
6463biimpac 503 . . . . . . 7 ((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩) → ⟨“𝐴(𝑊‘1)𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))
6562, 64jca 554 . . . . . 6 ((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ ⟨“𝐴(𝑊‘1)𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))
6665adantr 481 . . . . 5 (((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩) ∧ (𝑊‘1) ∈ 𝑉) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ ⟨“𝐴(𝑊‘1)𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))
6752, 61, 66rspcedvd 3307 . . . 4 (((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩) ∧ (𝑊‘1) ∈ 𝑉) → ∃𝑏𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))
6851, 67syl 17 . . 3 (((𝐺𝑈𝐴𝑉𝐶𝑉) ∧ 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → ∃𝑏𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))
6968ex 450 . 2 ((𝐺𝑈𝐴𝑉𝐶𝑉) → (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → ∃𝑏𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))))
70 eleq1 2692 . . . . . 6 (⟨“𝐴𝑏𝐶”⟩ = 𝑊 → (⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))
7170eqcoms 2634 . . . . 5 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ → (⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))
7271biimpa 501 . . . 4 ((𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))
7372a1i 11 . . 3 ((𝐺𝑈𝐴𝑉𝐶𝑉) → ((𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))
7473rexlimdvw 3032 . 2 ((𝐺𝑈𝐴𝑉𝐶𝑉) → (∃𝑏𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))
7569, 74impbid 202 1 ((𝐺𝑈𝐴𝑉𝐶𝑉) → (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ∃𝑏𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1992  ∃wrex 2913  {ctp 4157  ‘cfv 5850  (class class class)co 6605  0cc0 9881  1c1 9882   + caddc 9884  2c2 11015  3c3 11016  ..^cfzo 12403  #chash 13054  Word cword 13225  ⟨“cs3 13519  Vtxcvtx 25769   WWalksN cwwlksn 26581   WWalksNOn cwwlksnon 26582 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-er 7688  df-map 7805  df-pm 7806  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-card 8710  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-2 11024  df-3 11025  df-n0 11238  df-z 11323  df-uz 11632  df-fz 12266  df-fzo 12404  df-hash 13055  df-word 13233  df-concat 13235  df-s1 13236  df-s2 13525  df-s3 13526  df-wwlks 26585  df-wwlksn 26586  df-wwlksnon 26587 This theorem is referenced by:  elwwlks2on  26714  frgr2wwlk1  27046
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