Theorem nb3grpr 29417

Description: The neighbors of a vertex in a simple graph with three elements are an unordered pair of the other vertices iff all vertices are connected with each other. (Contributed by Alexander van der Vekens, 18-Oct-2017.) (Revised by AV, 28-Oct-2020.)

Hypotheses

Ref	Expression
nb3grpr.v	⊢ 𝑉 = (Vtx‘𝐺)
nb3grpr.e	⊢ 𝐸 = (Edg‘𝐺)
nb3grpr.g	⊢ (𝜑 → 𝐺 ∈ USGraph)
nb3grpr.t	⊢ (𝜑 → 𝑉 = {𝐴, 𝐵, 𝐶})
nb3grpr.s	⊢ (𝜑 → (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍))
nb3grpr.n	⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶))

Assertion

Ref	Expression
nb3grpr	⊢ (𝜑 → (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸) ↔ ∀𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝑥) = {𝑦, 𝑧}))

Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑦,𝐸   𝑥,𝐺,𝑦,𝑧   𝑥,𝑉,𝑦,𝑧   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑧)   𝐸(𝑥,𝑧)   𝑋(𝑥,𝑦,𝑧)   𝑌(𝑥,𝑦,𝑧)   𝑍(𝑥,𝑦,𝑧)

Proof of Theorem nb3grpr

Step	Hyp	Ref	Expression
1		id 22	. . . . . 6 ⊢ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸) → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸))
2		prcom 4757	. . . . . . . . . 10 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴}
3	2	eleq1i 2835	. . . . . . . . 9 ⊢ ({𝐴, 𝐵} ∈ 𝐸 ↔ {𝐵, 𝐴} ∈ 𝐸)
4		prcom 4757	. . . . . . . . . 10 ⊢ {𝐵, 𝐶} = {𝐶, 𝐵}
5	4	eleq1i 2835	. . . . . . . . 9 ⊢ ({𝐵, 𝐶} ∈ 𝐸 ↔ {𝐶, 𝐵} ∈ 𝐸)
6		prcom 4757	. . . . . . . . . 10 ⊢ {𝐶, 𝐴} = {𝐴, 𝐶}
7	6	eleq1i 2835	. . . . . . . . 9 ⊢ ({𝐶, 𝐴} ∈ 𝐸 ↔ {𝐴, 𝐶} ∈ 𝐸)
8	3, 5, 7	3anbi123i 1155	. . . . . . . 8 ⊢ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸) ↔ ({𝐵, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))
9		3anrot 1100	. . . . . . . 8 ⊢ (({𝐴, 𝐶} ∈ 𝐸 ∧ {𝐵, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸) ↔ ({𝐵, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))
10	8, 9	bitr4i 278	. . . . . . 7 ⊢ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸) ↔ ({𝐴, 𝐶} ∈ 𝐸 ∧ {𝐵, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸))
11	10	a1i 11	. . . . . 6 ⊢ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸) → (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸) ↔ ({𝐴, 𝐶} ∈ 𝐸 ∧ {𝐵, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
12	1, 11	biadanii 821	. . . . 5 ⊢ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸) ↔ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸) ∧ ({𝐴, 𝐶} ∈ 𝐸 ∧ {𝐵, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
13		an6 1445	. . . . 5 ⊢ ((({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸) ∧ ({𝐴, 𝐶} ∈ 𝐸 ∧ {𝐵, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)) ↔ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸) ∧ ({𝐵, 𝐶} ∈ 𝐸 ∧ {𝐵, 𝐴} ∈ 𝐸) ∧ ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
14	12, 13	bitri 275	. . . 4 ⊢ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸) ↔ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸) ∧ ({𝐵, 𝐶} ∈ 𝐸 ∧ {𝐵, 𝐴} ∈ 𝐸) ∧ ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
15	14	a1i 11	. . 3 ⊢ (𝜑 → (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸) ↔ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸) ∧ ({𝐵, 𝐶} ∈ 𝐸 ∧ {𝐵, 𝐴} ∈ 𝐸) ∧ ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸))))
16		nb3grpr.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
17		nb3grpr.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
18		nb3grpr.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ USGraph)
19		nb3grpr.t	. . . . 5 ⊢ (𝜑 → 𝑉 = {𝐴, 𝐵, 𝐶})
20		nb3grpr.s	. . . . 5 ⊢ (𝜑 → (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍))
21	16, 17, 18, 19, 20	nb3grprlem1 29415	. . . 4 ⊢ (𝜑 → ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)))
22		tprot 4774	. . . . . 6 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
23	19, 22	eqtrdi 2796	. . . . 5 ⊢ (𝜑 → 𝑉 = {𝐵, 𝐶, 𝐴})
24		3anrot 1100	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ↔ (𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍 ∧ 𝐴 ∈ 𝑋))
25	20, 24	sylib 218	. . . . 5 ⊢ (𝜑 → (𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍 ∧ 𝐴 ∈ 𝑋))
26	16, 17, 18, 23, 25	nb3grprlem1 29415	. . . 4 ⊢ (𝜑 → ((𝐺 NeighbVtx 𝐵) = {𝐶, 𝐴} ↔ ({𝐵, 𝐶} ∈ 𝐸 ∧ {𝐵, 𝐴} ∈ 𝐸)))
27		tprot 4774	. . . . . 6 ⊢ {𝐶, 𝐴, 𝐵} = {𝐴, 𝐵, 𝐶}
28	19, 27	eqtr4di 2798	. . . . 5 ⊢ (𝜑 → 𝑉 = {𝐶, 𝐴, 𝐵})
29		3anrot 1100	. . . . . 6 ⊢ ((𝐶 ∈ 𝑍 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍))
30	20, 29	sylibr 234	. . . . 5 ⊢ (𝜑 → (𝐶 ∈ 𝑍 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌))
31	16, 17, 18, 28, 30	nb3grprlem1 29415	. . . 4 ⊢ (𝜑 → ((𝐺 NeighbVtx 𝐶) = {𝐴, 𝐵} ↔ ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
32	21, 26, 31	3anbi123d 1436	. . 3 ⊢ (𝜑 → (((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∧ (𝐺 NeighbVtx 𝐵) = {𝐶, 𝐴} ∧ (𝐺 NeighbVtx 𝐶) = {𝐴, 𝐵}) ↔ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸) ∧ ({𝐵, 𝐶} ∈ 𝐸 ∧ {𝐵, 𝐴} ∈ 𝐸) ∧ ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸))))
33		nb3grpr.n	. . . . 5 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶))
34	16, 17, 18, 19, 20, 33	nb3grprlem2 29416	. . . 4 ⊢ (𝜑 → ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ↔ ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝐴) = {𝑦, 𝑧}))
35		necom 3000	. . . . . . . 8 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴)
36		necom 3000	. . . . . . . 8 ⊢ (𝐴 ≠ 𝐶 ↔ 𝐶 ≠ 𝐴)
37		biid 261	. . . . . . . 8 ⊢ (𝐵 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶)
38	35, 36, 37	3anbi123i 1155	. . . . . . 7 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ↔ (𝐵 ≠ 𝐴 ∧ 𝐶 ≠ 𝐴 ∧ 𝐵 ≠ 𝐶))
39		3anrot 1100	. . . . . . 7 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐴 ∧ 𝐶 ≠ 𝐴) ↔ (𝐵 ≠ 𝐴 ∧ 𝐶 ≠ 𝐴 ∧ 𝐵 ≠ 𝐶))
40	38, 39	bitr4i 278	. . . . . 6 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ↔ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐴 ∧ 𝐶 ≠ 𝐴))
41	33, 40	sylib 218	. . . . 5 ⊢ (𝜑 → (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐴 ∧ 𝐶 ≠ 𝐴))
42	16, 17, 18, 23, 25, 41	nb3grprlem2 29416	. . . 4 ⊢ (𝜑 → ((𝐺 NeighbVtx 𝐵) = {𝐶, 𝐴} ↔ ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝐵) = {𝑦, 𝑧}))
43		3anrot 1100	. . . . . . 7 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ↔ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ 𝐴 ≠ 𝐵))
44		necom 3000	. . . . . . . 8 ⊢ (𝐵 ≠ 𝐶 ↔ 𝐶 ≠ 𝐵)
45		biid 261	. . . . . . . 8 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐴 ≠ 𝐵)
46	36, 44, 45	3anbi123i 1155	. . . . . . 7 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ 𝐴 ≠ 𝐵) ↔ (𝐶 ≠ 𝐴 ∧ 𝐶 ≠ 𝐵 ∧ 𝐴 ≠ 𝐵))
47	43, 46	bitri 275	. . . . . 6 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ↔ (𝐶 ≠ 𝐴 ∧ 𝐶 ≠ 𝐵 ∧ 𝐴 ≠ 𝐵))
48	33, 47	sylib 218	. . . . 5 ⊢ (𝜑 → (𝐶 ≠ 𝐴 ∧ 𝐶 ≠ 𝐵 ∧ 𝐴 ≠ 𝐵))
49	16, 17, 18, 28, 30, 48	nb3grprlem2 29416	. . . 4 ⊢ (𝜑 → ((𝐺 NeighbVtx 𝐶) = {𝐴, 𝐵} ↔ ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝐶) = {𝑦, 𝑧}))
50	34, 42, 49	3anbi123d 1436	. . 3 ⊢ (𝜑 → (((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∧ (𝐺 NeighbVtx 𝐵) = {𝐶, 𝐴} ∧ (𝐺 NeighbVtx 𝐶) = {𝐴, 𝐵}) ↔ (∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝐴) = {𝑦, 𝑧} ∧ ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝐵) = {𝑦, 𝑧} ∧ ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝐶) = {𝑦, 𝑧})))
51	15, 32, 50	3bitr2d 307	. 2 ⊢ (𝜑 → (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸) ↔ (∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝐴) = {𝑦, 𝑧} ∧ ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝐵) = {𝑦, 𝑧} ∧ ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝐶) = {𝑦, 𝑧})))
52		oveq2 7456	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝐴))
53	52	eqeq1d 2742	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝐺 NeighbVtx 𝑥) = {𝑦, 𝑧} ↔ (𝐺 NeighbVtx 𝐴) = {𝑦, 𝑧}))
54	53	2rexbidv 3228	. . . 4 ⊢ (𝑥 = 𝐴 → (∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝑥) = {𝑦, 𝑧} ↔ ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝐴) = {𝑦, 𝑧}))
55		oveq2 7456	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝐵))
56	55	eqeq1d 2742	. . . . 5 ⊢ (𝑥 = 𝐵 → ((𝐺 NeighbVtx 𝑥) = {𝑦, 𝑧} ↔ (𝐺 NeighbVtx 𝐵) = {𝑦, 𝑧}))
57	56	2rexbidv 3228	. . . 4 ⊢ (𝑥 = 𝐵 → (∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝑥) = {𝑦, 𝑧} ↔ ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝐵) = {𝑦, 𝑧}))
58		oveq2 7456	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝐶))
59	58	eqeq1d 2742	. . . . 5 ⊢ (𝑥 = 𝐶 → ((𝐺 NeighbVtx 𝑥) = {𝑦, 𝑧} ↔ (𝐺 NeighbVtx 𝐶) = {𝑦, 𝑧}))
60	59	2rexbidv 3228	. . . 4 ⊢ (𝑥 = 𝐶 → (∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝑥) = {𝑦, 𝑧} ↔ ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝐶) = {𝑦, 𝑧}))
61	54, 57, 60	raltpg 4723	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝑥) = {𝑦, 𝑧} ↔ (∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝐴) = {𝑦, 𝑧} ∧ ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝐵) = {𝑦, 𝑧} ∧ ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝐶) = {𝑦, 𝑧})))
62	20, 61	syl 17	. 2 ⊢ (𝜑 → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝑥) = {𝑦, 𝑧} ↔ (∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝐴) = {𝑦, 𝑧} ∧ ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝐵) = {𝑦, 𝑧} ∧ ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝐶) = {𝑦, 𝑧})))
63		raleq 3331	. . . 4 ⊢ (𝑉 = {𝐴, 𝐵, 𝐶} → (∀𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝑥) = {𝑦, 𝑧} ↔ ∀𝑥 ∈ {𝐴, 𝐵, 𝐶}∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝑥) = {𝑦, 𝑧}))
64	63	bicomd 223	. . 3 ⊢ (𝑉 = {𝐴, 𝐵, 𝐶} → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝑥) = {𝑦, 𝑧} ↔ ∀𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝑥) = {𝑦, 𝑧}))
65	19, 64	syl 17	. 2 ⊢ (𝜑 → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝑥) = {𝑦, 𝑧} ↔ ∀𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝑥) = {𝑦, 𝑧}))
66	51, 62, 65	3bitr2d 307	1 ⊢ (𝜑 → (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸) ↔ ∀𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝑥) = {𝑦, 𝑧}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  ∃wrex 3076   ∖ cdif 3973  {csn 4648  {cpr 4650  {ctp 4652  ‘cfv 6573  (class class class)co 7448  Vtxcvtx 29031  Edgcedg 29082  USGraphcusgr 29184   NeighbVtx cnbgr 29367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-oadd 8526  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-dju 9970  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-n0 12554  df-xnn0 12626  df-z 12640  df-uz 12904  df-fz 13568  df-hash 14380  df-edg 29083  df-upgr 29117  df-umgr 29118  df-usgr 29186  df-nbgr 29368

This theorem is referenced by:  cusgr3vnbpr  29471