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Theorem elwwlks2ons3OLD 27076
 Description: Obsolete version of elwwlks2ons3 27075 as of 13-Mar-2022. (Contributed by Alexander van der Vekens, 15-Feb-2018.) (Revised by AV, 12-May-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
wwlks2onv.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
elwwlks2ons3OLD ((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   𝑉,𝑏   𝑊,𝑏   𝑈,𝑏

Proof of Theorem elwwlks2ons3OLD
StepHypRef Expression
1 simpr 479 . . . . 5 (((𝐺𝑈𝐴𝑉𝐶𝑉) ∧ 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))
2 wwlks2onv.v . . . . . . . . 9 𝑉 = (Vtx‘𝐺)
32wwlknonOLD 26965 . . . . . . . 8 ((𝐴𝑉𝐶𝑉) → (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ (𝑊 ∈ (2 WWalksN 𝐺) ∧ (𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶)))
433adant1 1125 . . . . . . 7 ((𝐺𝑈𝐴𝑉𝐶𝑉) → (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ (𝑊 ∈ (2 WWalksN 𝐺) ∧ (𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶)))
5 wwlknbp2OLD 26949 . . . . . . . . . 10 (𝑊 ∈ (2 WWalksN 𝐺) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (2 + 1)))
6 2p1e3 11343 . . . . . . . . . . . 12 (2 + 1) = 3
76eqeq2i 2772 . . . . . . . . . . 11 ((♯‘𝑊) = (2 + 1) ↔ (♯‘𝑊) = 3)
8 1ex 10227 . . . . . . . . . . . . . . . . 17 1 ∈ V
98tpid2 4448 . . . . . . . . . . . . . . . 16 1 ∈ {0, 1, 2}
10 oveq2 6821 . . . . . . . . . . . . . . . . 17 ((♯‘𝑊) = 3 → (0..^(♯‘𝑊)) = (0..^3))
11 fzo0to3tp 12748 . . . . . . . . . . . . . . . . 17 (0..^3) = {0, 1, 2}
1210, 11syl6eq 2810 . . . . . . . . . . . . . . . 16 ((♯‘𝑊) = 3 → (0..^(♯‘𝑊)) = {0, 1, 2})
139, 12syl5eleqr 2846 . . . . . . . . . . . . . . 15 ((♯‘𝑊) = 3 → 1 ∈ (0..^(♯‘𝑊)))
14 wrdsymbcl 13504 . . . . . . . . . . . . . . 15 ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 1 ∈ (0..^(♯‘𝑊))) → (𝑊‘1) ∈ (Vtx‘𝐺))
1513, 14sylan2 492 . . . . . . . . . . . . . 14 ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 3) → (𝑊‘1) ∈ (Vtx‘𝐺))
16153ad2ant1 1128 . . . . . . . . . . . . 13 (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐺𝑈𝐴𝑉𝐶𝑉)) → (𝑊‘1) ∈ (Vtx‘𝐺))
17 simpr 479 . . . . . . . . . . . . . . . . 17 ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 3) → (♯‘𝑊) = 3)
18173ad2ant1 1128 . . . . . . . . . . . . . . . 16 (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐺𝑈𝐴𝑉𝐶𝑉)) → (♯‘𝑊) = 3)
1918adantr 472 . . . . . . . . . . . . . . 15 ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐺𝑈𝐴𝑉𝐶𝑉)) ∧ (𝑊‘1) ∈ (Vtx‘𝐺)) → (♯‘𝑊) = 3)
20 simpl 474 . . . . . . . . . . . . . . . . . 18 (((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) → (𝑊‘0) = 𝐴)
21 eqidd 2761 . . . . . . . . . . . . . . . . . 18 (((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) → (𝑊‘1) = (𝑊‘1))
22 simpr 479 . . . . . . . . . . . . . . . . . 18 (((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) → (𝑊‘2) = 𝐶)
2320, 21, 223jca 1123 . . . . . . . . . . . . . . . . 17 (((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) → ((𝑊‘0) = 𝐴 ∧ (𝑊‘1) = (𝑊‘1) ∧ (𝑊‘2) = 𝐶))
24233ad2ant2 1129 . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . 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 2769 . . . . . . . . . . . . . . . . . . . . . 22 (Vtx‘𝐺) = 𝑉
2726wrdeqi 13514 . . . . . . . . . . . . . . . . . . . . 21 Word (Vtx‘𝐺) = Word 𝑉
2827eleq2i 2831 . . . . . . . . . . . . . . . . . . . 20 (𝑊 ∈ Word (Vtx‘𝐺) ↔ 𝑊 ∈ Word 𝑉)
2928biimpi 206 . . . . . . . . . . . . . . . . . . 19 (𝑊 ∈ Word (Vtx‘𝐺) → 𝑊 ∈ Word 𝑉)
3029adantr 472 . . . . . . . . . . . . . . . . . 18 ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 3) → 𝑊 ∈ Word 𝑉)
31303ad2ant1 1128 . . . . . . . . . . . . . . . . 17 (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐺𝑈𝐴𝑉𝐶𝑉)) → 𝑊 ∈ Word 𝑉)
3231adantr 472 . . . . . . . . . . . . . . . 16 ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐺𝑈𝐴𝑉𝐶𝑉)) ∧ (𝑊‘1) ∈ (Vtx‘𝐺)) → 𝑊 ∈ Word 𝑉)
33 simpl32 1329 . . . . . . . . . . . . . . . 16 ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐺𝑈𝐴𝑉𝐶𝑉)) ∧ (𝑊‘1) ∈ (Vtx‘𝐺)) → 𝐴𝑉)
3426eleq2i 2831 . . . . . . . . . . . . . . . . . 18 ((𝑊‘1) ∈ (Vtx‘𝐺) ↔ (𝑊‘1) ∈ 𝑉)
3534biimpi 206 . . . . . . . . . . . . . . . . 17 ((𝑊‘1) ∈ (Vtx‘𝐺) → (𝑊‘1) ∈ 𝑉)
3635adantl 473 . . . . . . . . . . . . . . . 16 ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐺𝑈𝐴𝑉𝐶𝑉)) ∧ (𝑊‘1) ∈ (Vtx‘𝐺)) → (𝑊‘1) ∈ 𝑉)
37 simpl33 1331 . . . . . . . . . . . . . . . 16 ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐺𝑈𝐴𝑉𝐶𝑉)) ∧ (𝑊‘1) ∈ (Vtx‘𝐺)) → 𝐶𝑉)
38 eqwrds3 13905 . . . . . . . . . . . . . . . 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 1479 . . . . . . . . . . . . . . 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 995 . . . . . . . . . . . . . 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 555 . . . . . . . . . . . . 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 705 . . . . . . . . . . . 12 (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐺𝑈𝐴𝑉𝐶𝑉)) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ (𝑊‘1) ∈ 𝑉))
43423exp 1113 . . . . . . . . . . 11 ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 3) → (((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) → ((𝐺𝑈𝐴𝑉𝐶𝑉) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ (𝑊‘1) ∈ 𝑉))))
447, 43sylan2b 493 . . . . . . . . . 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 1109 . . . . . . . 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 444 . . . . 5 (((𝐺𝑈𝐴𝑉𝐶𝑉) ∧ 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ (𝑊‘1) ∈ 𝑉))
50 anass 684 . . . . 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 701 . . . 4 (((𝐺𝑈𝐴𝑉𝐶𝑉) ∧ 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → ((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩) ∧ (𝑊‘1) ∈ 𝑉))
52 simpr 479 . . . . 5 (((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩) ∧ (𝑊‘1) ∈ 𝑉) → (𝑊‘1) ∈ 𝑉)
53 eqidd 2761 . . . . . . . 8 (𝑏 = (𝑊‘1) → 𝐴 = 𝐴)
54 id 22 . . . . . . . 8 (𝑏 = (𝑊‘1) → 𝑏 = (𝑊‘1))
55 eqidd 2761 . . . . . . . 8 (𝑏 = (𝑊‘1) → 𝐶 = 𝐶)
5653, 54, 55s3eqd 13809 . . . . . . 7 (𝑏 = (𝑊‘1) → ⟨“𝐴𝑏𝐶”⟩ = ⟨“𝐴(𝑊‘1)𝐶”⟩)
57 eqeq2 2771 . . . . . . . 8 (⟨“𝐴𝑏𝐶”⟩ = ⟨“𝐴(𝑊‘1)𝐶”⟩ → (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ↔ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩))
58 eleq1 2827 . . . . . . . 8 (⟨“𝐴𝑏𝐶”⟩ = ⟨“𝐴(𝑊‘1)𝐶”⟩ → (⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ⟨“𝐴(𝑊‘1)𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))
5957, 58anbi12d 749 . . . . . . 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 473 . . . . 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 479 . . . . . . 7 ((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩) → 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩)
63 eleq1 2827 . . . . . . . 8 (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ → (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ⟨“𝐴(𝑊‘1)𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))
6463biimpac 504 . . . . . . 7 ((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩) → ⟨“𝐴(𝑊‘1)𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))
6562, 64jca 555 . . . . . 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 472 . . . . 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 3456 . . . 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 449 . 2 ((𝐺𝑈𝐴𝑉𝐶𝑉) → (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → ∃𝑏𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))))
70 eleq1 2827 . . . . . 6 (⟨“𝐴𝑏𝐶”⟩ = 𝑊 → (⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))
7170eqcoms 2768 . . . . 5 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ → (⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))
7271biimpa 502 . . . 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 3172 . 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 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139  ∃wrex 3051  {ctp 4325  ‘cfv 6049  (class class class)co 6813  0cc0 10128  1c1 10129   + caddc 10131  2c2 11262  3c3 11263  ..^cfzo 12659  ♯chash 13311  Word cword 13477  ⟨“cs3 13787  Vtxcvtx 26073   WWalksN cwwlksn 26929   WWalksNOn cwwlksnon 26930 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-map 8025  df-pm 8026  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-card 8955  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-3 11272  df-n0 11485  df-z 11570  df-uz 11880  df-fz 12520  df-fzo 12660  df-hash 13312  df-word 13485  df-concat 13487  df-s1 13488  df-s2 13793  df-s3 13794  df-wwlks 26933  df-wwlksn 26934  df-wwlksnon 26935 This theorem is referenced by: (None)
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