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Theorem nb3grapr 25982
 Description: The neighbors of a vertex in a graph with three elements are an unordered pair of the other vertices if and only if all vertices are connected with each other. (Contributed by Alexander van der Vekens, 18-Oct-2017.)
Assertion
Ref Expression
nb3grapr (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ↔ ∀𝑥𝑉𝑦𝑉𝑧 ∈ (𝑉 ∖ {𝑦})(⟨𝑉, 𝐸⟩ Neighbors 𝑥) = {𝑦, 𝑧}))
Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑥,𝐸,𝑦,𝑧   𝑥,𝑉,𝑦,𝑧
Allowed substitution hints:   𝑋(𝑥,𝑦,𝑧)   𝑌(𝑥,𝑦,𝑧)   𝑍(𝑥,𝑦,𝑧)

Proof of Theorem nb3grapr
StepHypRef Expression
1 id 22 . . . . . 6 (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))
2 prcom 4211 . . . . . . . . . 10 {𝐴, 𝐵} = {𝐵, 𝐴}
32eleq1i 2679 . . . . . . . . 9 ({𝐴, 𝐵} ∈ ran 𝐸 ↔ {𝐵, 𝐴} ∈ ran 𝐸)
4 prcom 4211 . . . . . . . . . 10 {𝐵, 𝐶} = {𝐶, 𝐵}
54eleq1i 2679 . . . . . . . . 9 ({𝐵, 𝐶} ∈ ran 𝐸 ↔ {𝐶, 𝐵} ∈ ran 𝐸)
6 prcom 4211 . . . . . . . . . 10 {𝐶, 𝐴} = {𝐴, 𝐶}
76eleq1i 2679 . . . . . . . . 9 ({𝐶, 𝐴} ∈ ran 𝐸 ↔ {𝐴, 𝐶} ∈ ran 𝐸)
83, 5, 73anbi123i 1244 . . . . . . . 8 (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ↔ ({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸))
9 3anrot 1036 . . . . . . . 8 (({𝐴, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸) ↔ ({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸))
108, 9bitr4i 266 . . . . . . 7 (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ↔ ({𝐴, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸))
1110a1i 11 . . . . . 6 (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ↔ ({𝐴, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)))
121, 11biadan2 672 . . . . 5 (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ↔ (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ∧ ({𝐴, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)))
13 an6 1400 . . . . 5 ((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ∧ ({𝐴, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)) ↔ (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸) ∧ ({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸) ∧ ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)))
1412, 13bitri 263 . . . 4 (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ↔ (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸) ∧ ({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸) ∧ ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)))
1514a1i 11 . . 3 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ↔ (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸) ∧ ({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸) ∧ ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸))))
16 nb3graprlem1 25980 . . . . 5 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) → ((⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶} ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)))
17 3anrot 1036 . . . . . . 7 ((𝐴𝑋𝐵𝑌𝐶𝑍) ↔ (𝐵𝑌𝐶𝑍𝐴𝑋))
1817biimpi 205 . . . . . 6 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (𝐵𝑌𝐶𝑍𝐴𝑋))
19 tprot 4228 . . . . . . . . 9 {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
2019eqeq2i 2622 . . . . . . . 8 (𝑉 = {𝐴, 𝐵, 𝐶} ↔ 𝑉 = {𝐵, 𝐶, 𝐴})
2120biimpi 205 . . . . . . 7 (𝑉 = {𝐴, 𝐵, 𝐶} → 𝑉 = {𝐵, 𝐶, 𝐴})
2221anim1i 590 . . . . . 6 ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸) → (𝑉 = {𝐵, 𝐶, 𝐴} ∧ 𝑉 USGrph 𝐸))
23 nb3graprlem1 25980 . . . . . 6 (((𝐵𝑌𝐶𝑍𝐴𝑋) ∧ (𝑉 = {𝐵, 𝐶, 𝐴} ∧ 𝑉 USGrph 𝐸)) → ((⟨𝑉, 𝐸⟩ Neighbors 𝐵) = {𝐶, 𝐴} ↔ ({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸)))
2418, 22, 23syl2an 493 . . . . 5 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) → ((⟨𝑉, 𝐸⟩ Neighbors 𝐵) = {𝐶, 𝐴} ↔ ({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸)))
25 3anrot 1036 . . . . . . 7 ((𝐶𝑍𝐴𝑋𝐵𝑌) ↔ (𝐴𝑋𝐵𝑌𝐶𝑍))
2625biimpri 217 . . . . . 6 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (𝐶𝑍𝐴𝑋𝐵𝑌))
27 tprot 4228 . . . . . . . . . 10 {𝐶, 𝐴, 𝐵} = {𝐴, 𝐵, 𝐶}
2827eqcomi 2619 . . . . . . . . 9 {𝐴, 𝐵, 𝐶} = {𝐶, 𝐴, 𝐵}
2928eqeq2i 2622 . . . . . . . 8 (𝑉 = {𝐴, 𝐵, 𝐶} ↔ 𝑉 = {𝐶, 𝐴, 𝐵})
3029biimpi 205 . . . . . . 7 (𝑉 = {𝐴, 𝐵, 𝐶} → 𝑉 = {𝐶, 𝐴, 𝐵})
3130anim1i 590 . . . . . 6 ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸) → (𝑉 = {𝐶, 𝐴, 𝐵} ∧ 𝑉 USGrph 𝐸))
32 nb3graprlem1 25980 . . . . . 6 (((𝐶𝑍𝐴𝑋𝐵𝑌) ∧ (𝑉 = {𝐶, 𝐴, 𝐵} ∧ 𝑉 USGrph 𝐸)) → ((⟨𝑉, 𝐸⟩ Neighbors 𝐶) = {𝐴, 𝐵} ↔ ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)))
3326, 31, 32syl2an 493 . . . . 5 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) → ((⟨𝑉, 𝐸⟩ Neighbors 𝐶) = {𝐴, 𝐵} ↔ ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)))
3416, 24, 333anbi123d 1391 . . . 4 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) → (((⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶} ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐵) = {𝐶, 𝐴} ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐶) = {𝐴, 𝐵}) ↔ (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸) ∧ ({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸) ∧ ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸))))
35343adant3 1074 . . 3 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (((⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶} ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐵) = {𝐶, 𝐴} ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐶) = {𝐴, 𝐵}) ↔ (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸) ∧ ({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸) ∧ ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸))))
36 nb3graprlem2 25981 . . . 4 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ((⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶} ↔ ∃𝑦𝑉𝑧 ∈ (𝑉 ∖ {𝑦})(⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝑦, 𝑧}))
3720anbi1i 727 . . . . 5 ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸) ↔ (𝑉 = {𝐵, 𝐶, 𝐴} ∧ 𝑉 USGrph 𝐸))
38 necom 2835 . . . . . . 7 (𝐴𝐵𝐵𝐴)
39 necom 2835 . . . . . . 7 (𝐴𝐶𝐶𝐴)
40 biid 250 . . . . . . 7 (𝐵𝐶𝐵𝐶)
4138, 39, 403anbi123i 1244 . . . . . 6 ((𝐴𝐵𝐴𝐶𝐵𝐶) ↔ (𝐵𝐴𝐶𝐴𝐵𝐶))
42 3anrot 1036 . . . . . 6 ((𝐵𝐶𝐵𝐴𝐶𝐴) ↔ (𝐵𝐴𝐶𝐴𝐵𝐶))
4341, 42bitr4i 266 . . . . 5 ((𝐴𝐵𝐴𝐶𝐵𝐶) ↔ (𝐵𝐶𝐵𝐴𝐶𝐴))
44 nb3graprlem2 25981 . . . . 5 (((𝐵𝑌𝐶𝑍𝐴𝑋) ∧ (𝑉 = {𝐵, 𝐶, 𝐴} ∧ 𝑉 USGrph 𝐸) ∧ (𝐵𝐶𝐵𝐴𝐶𝐴)) → ((⟨𝑉, 𝐸⟩ Neighbors 𝐵) = {𝐶, 𝐴} ↔ ∃𝑦𝑉𝑧 ∈ (𝑉 ∖ {𝑦})(⟨𝑉, 𝐸⟩ Neighbors 𝐵) = {𝑦, 𝑧}))
4517, 37, 43, 44syl3anb 1361 . . . 4 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ((⟨𝑉, 𝐸⟩ Neighbors 𝐵) = {𝐶, 𝐴} ↔ ∃𝑦𝑉𝑧 ∈ (𝑉 ∖ {𝑦})(⟨𝑉, 𝐸⟩ Neighbors 𝐵) = {𝑦, 𝑧}))
46 id 22 . . . . . . 7 (𝑉 = {𝐴, 𝐵, 𝐶} → 𝑉 = {𝐴, 𝐵, 𝐶})
4746, 28syl6eq 2660 . . . . . 6 (𝑉 = {𝐴, 𝐵, 𝐶} → 𝑉 = {𝐶, 𝐴, 𝐵})
4847anim1i 590 . . . . 5 ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸) → (𝑉 = {𝐶, 𝐴, 𝐵} ∧ 𝑉 USGrph 𝐸))
49 3anrot 1036 . . . . . . 7 ((𝐴𝐵𝐴𝐶𝐵𝐶) ↔ (𝐴𝐶𝐵𝐶𝐴𝐵))
50 necom 2835 . . . . . . . 8 (𝐵𝐶𝐶𝐵)
51 biid 250 . . . . . . . 8 (𝐴𝐵𝐴𝐵)
5239, 50, 513anbi123i 1244 . . . . . . 7 ((𝐴𝐶𝐵𝐶𝐴𝐵) ↔ (𝐶𝐴𝐶𝐵𝐴𝐵))
5349, 52bitri 263 . . . . . 6 ((𝐴𝐵𝐴𝐶𝐵𝐶) ↔ (𝐶𝐴𝐶𝐵𝐴𝐵))
5453biimpi 205 . . . . 5 ((𝐴𝐵𝐴𝐶𝐵𝐶) → (𝐶𝐴𝐶𝐵𝐴𝐵))
55 nb3graprlem2 25981 . . . . 5 (((𝐶𝑍𝐴𝑋𝐵𝑌) ∧ (𝑉 = {𝐶, 𝐴, 𝐵} ∧ 𝑉 USGrph 𝐸) ∧ (𝐶𝐴𝐶𝐵𝐴𝐵)) → ((⟨𝑉, 𝐸⟩ Neighbors 𝐶) = {𝐴, 𝐵} ↔ ∃𝑦𝑉𝑧 ∈ (𝑉 ∖ {𝑦})(⟨𝑉, 𝐸⟩ Neighbors 𝐶) = {𝑦, 𝑧}))
5626, 48, 54, 55syl3an 1360 . . . 4 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ((⟨𝑉, 𝐸⟩ Neighbors 𝐶) = {𝐴, 𝐵} ↔ ∃𝑦𝑉𝑧 ∈ (𝑉 ∖ {𝑦})(⟨𝑉, 𝐸⟩ Neighbors 𝐶) = {𝑦, 𝑧}))
5736, 45, 563anbi123d 1391 . . 3 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (((⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶} ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐵) = {𝐶, 𝐴} ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐶) = {𝐴, 𝐵}) ↔ (∃𝑦𝑉𝑧 ∈ (𝑉 ∖ {𝑦})(⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝑦, 𝑧} ∧ ∃𝑦𝑉𝑧 ∈ (𝑉 ∖ {𝑦})(⟨𝑉, 𝐸⟩ Neighbors 𝐵) = {𝑦, 𝑧} ∧ ∃𝑦𝑉𝑧 ∈ (𝑉 ∖ {𝑦})(⟨𝑉, 𝐸⟩ Neighbors 𝐶) = {𝑦, 𝑧})))
5815, 35, 573bitr2d 295 . 2 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ↔ (∃𝑦𝑉𝑧 ∈ (𝑉 ∖ {𝑦})(⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝑦, 𝑧} ∧ ∃𝑦𝑉𝑧 ∈ (𝑉 ∖ {𝑦})(⟨𝑉, 𝐸⟩ Neighbors 𝐵) = {𝑦, 𝑧} ∧ ∃𝑦𝑉𝑧 ∈ (𝑉 ∖ {𝑦})(⟨𝑉, 𝐸⟩ Neighbors 𝐶) = {𝑦, 𝑧})))
59 oveq2 6557 . . . . . 6 (𝑥 = 𝐴 → (⟨𝑉, 𝐸⟩ Neighbors 𝑥) = (⟨𝑉, 𝐸⟩ Neighbors 𝐴))
6059eqeq1d 2612 . . . . 5 (𝑥 = 𝐴 → ((⟨𝑉, 𝐸⟩ Neighbors 𝑥) = {𝑦, 𝑧} ↔ (⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝑦, 𝑧}))
61602rexbidv 3039 . . . 4 (𝑥 = 𝐴 → (∃𝑦𝑉𝑧 ∈ (𝑉 ∖ {𝑦})(⟨𝑉, 𝐸⟩ Neighbors 𝑥) = {𝑦, 𝑧} ↔ ∃𝑦𝑉𝑧 ∈ (𝑉 ∖ {𝑦})(⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝑦, 𝑧}))
62 oveq2 6557 . . . . . 6 (𝑥 = 𝐵 → (⟨𝑉, 𝐸⟩ Neighbors 𝑥) = (⟨𝑉, 𝐸⟩ Neighbors 𝐵))
6362eqeq1d 2612 . . . . 5 (𝑥 = 𝐵 → ((⟨𝑉, 𝐸⟩ Neighbors 𝑥) = {𝑦, 𝑧} ↔ (⟨𝑉, 𝐸⟩ Neighbors 𝐵) = {𝑦, 𝑧}))
64632rexbidv 3039 . . . 4 (𝑥 = 𝐵 → (∃𝑦𝑉𝑧 ∈ (𝑉 ∖ {𝑦})(⟨𝑉, 𝐸⟩ Neighbors 𝑥) = {𝑦, 𝑧} ↔ ∃𝑦𝑉𝑧 ∈ (𝑉 ∖ {𝑦})(⟨𝑉, 𝐸⟩ Neighbors 𝐵) = {𝑦, 𝑧}))
65 oveq2 6557 . . . . . 6 (𝑥 = 𝐶 → (⟨𝑉, 𝐸⟩ Neighbors 𝑥) = (⟨𝑉, 𝐸⟩ Neighbors 𝐶))
6665eqeq1d 2612 . . . . 5 (𝑥 = 𝐶 → ((⟨𝑉, 𝐸⟩ Neighbors 𝑥) = {𝑦, 𝑧} ↔ (⟨𝑉, 𝐸⟩ Neighbors 𝐶) = {𝑦, 𝑧}))
67662rexbidv 3039 . . . 4 (𝑥 = 𝐶 → (∃𝑦𝑉𝑧 ∈ (𝑉 ∖ {𝑦})(⟨𝑉, 𝐸⟩ Neighbors 𝑥) = {𝑦, 𝑧} ↔ ∃𝑦𝑉𝑧 ∈ (𝑉 ∖ {𝑦})(⟨𝑉, 𝐸⟩ Neighbors 𝐶) = {𝑦, 𝑧}))
6861, 64, 67raltpg 4183 . . 3 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}∃𝑦𝑉𝑧 ∈ (𝑉 ∖ {𝑦})(⟨𝑉, 𝐸⟩ Neighbors 𝑥) = {𝑦, 𝑧} ↔ (∃𝑦𝑉𝑧 ∈ (𝑉 ∖ {𝑦})(⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝑦, 𝑧} ∧ ∃𝑦𝑉𝑧 ∈ (𝑉 ∖ {𝑦})(⟨𝑉, 𝐸⟩ Neighbors 𝐵) = {𝑦, 𝑧} ∧ ∃𝑦𝑉𝑧 ∈ (𝑉 ∖ {𝑦})(⟨𝑉, 𝐸⟩ Neighbors 𝐶) = {𝑦, 𝑧})))
69683ad2ant1 1075 . 2 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}∃𝑦𝑉𝑧 ∈ (𝑉 ∖ {𝑦})(⟨𝑉, 𝐸⟩ Neighbors 𝑥) = {𝑦, 𝑧} ↔ (∃𝑦𝑉𝑧 ∈ (𝑉 ∖ {𝑦})(⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝑦, 𝑧} ∧ ∃𝑦𝑉𝑧 ∈ (𝑉 ∖ {𝑦})(⟨𝑉, 𝐸⟩ Neighbors 𝐵) = {𝑦, 𝑧} ∧ ∃𝑦𝑉𝑧 ∈ (𝑉 ∖ {𝑦})(⟨𝑉, 𝐸⟩ Neighbors 𝐶) = {𝑦, 𝑧})))
70 raleq 3115 . . . . 5 (𝑉 = {𝐴, 𝐵, 𝐶} → (∀𝑥𝑉𝑦𝑉𝑧 ∈ (𝑉 ∖ {𝑦})(⟨𝑉, 𝐸⟩ Neighbors 𝑥) = {𝑦, 𝑧} ↔ ∀𝑥 ∈ {𝐴, 𝐵, 𝐶}∃𝑦𝑉𝑧 ∈ (𝑉 ∖ {𝑦})(⟨𝑉, 𝐸⟩ Neighbors 𝑥) = {𝑦, 𝑧}))
7170bicomd 212 . . . 4 (𝑉 = {𝐴, 𝐵, 𝐶} → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}∃𝑦𝑉𝑧 ∈ (𝑉 ∖ {𝑦})(⟨𝑉, 𝐸⟩ Neighbors 𝑥) = {𝑦, 𝑧} ↔ ∀𝑥𝑉𝑦𝑉𝑧 ∈ (𝑉 ∖ {𝑦})(⟨𝑉, 𝐸⟩ Neighbors 𝑥) = {𝑦, 𝑧}))
7271adantr 480 . . 3 ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸) → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}∃𝑦𝑉𝑧 ∈ (𝑉 ∖ {𝑦})(⟨𝑉, 𝐸⟩ Neighbors 𝑥) = {𝑦, 𝑧} ↔ ∀𝑥𝑉𝑦𝑉𝑧 ∈ (𝑉 ∖ {𝑦})(⟨𝑉, 𝐸⟩ Neighbors 𝑥) = {𝑦, 𝑧}))
73723ad2ant2 1076 . 2 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}∃𝑦𝑉𝑧 ∈ (𝑉 ∖ {𝑦})(⟨𝑉, 𝐸⟩ Neighbors 𝑥) = {𝑦, 𝑧} ↔ ∀𝑥𝑉𝑦𝑉𝑧 ∈ (𝑉 ∖ {𝑦})(⟨𝑉, 𝐸⟩ Neighbors 𝑥) = {𝑦, 𝑧}))
7458, 69, 733bitr2d 295 1 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ↔ ∀𝑥𝑉𝑦𝑉𝑧 ∈ (𝑉 ∖ {𝑦})(⟨𝑉, 𝐸⟩ Neighbors 𝑥) = {𝑦, 𝑧}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897   ∖ cdif 3537  {csn 4125  {cpr 4127  {ctp 4129  ⟨cop 4131   class class class wbr 4583  ran crn 5039  (class class class)co 6549   USGrph cusg 25859   Neighbors cnbgra 25946 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-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-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-hash 12980  df-usgra 25862  df-nbgra 25949 This theorem is referenced by:  cusgra3vnbpr  25994
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