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Theorem 3vfriswmgr 27258
Description: Every friendship graph with three (different) vertices is a windmill graph. (Contributed by Alexander van der Vekens, 6-Oct-2017.) (Revised by AV, 31-Mar-2021.)
Hypotheses
Ref Expression
3vfriswmgr.v 𝑉 = (Vtx‘𝐺)
3vfriswmgr.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
3vfriswmgr (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸)))
Distinct variable groups:   𝑤,𝐴   𝑤,𝐵   𝑤,𝐶   𝑤,𝐸   𝑤,𝐺   𝑤,𝑉   𝑤,𝑋   𝑤,𝑌   𝐴,,𝑣,𝑤   𝐵,,𝑣   𝐶,,𝑣   ,𝐸,𝑣   ,𝑉,𝑣
Allowed substitution hints:   𝐺(𝑣,)   𝑋(𝑣,)   𝑌(𝑣,)   𝑍(𝑤,𝑣,)

Proof of Theorem 3vfriswmgr
StepHypRef Expression
1 frgrusgr 27240 . . . 4 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
2 3vfriswmgr.v . . . . . . . . . 10 𝑉 = (Vtx‘𝐺)
3 3vfriswmgr.e . . . . . . . . . 10 𝐸 = (Edg‘𝐺)
42, 3frgr3v 27255 . . . . . . . . 9 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → (𝐺 ∈ FriendGraph ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸))))
54exp4b 631 . . . . . . . 8 ((𝐴𝑋𝐵𝑌𝐶𝑍) → ((𝐴𝐵𝐴𝐶𝐵𝐶) → (𝑉 = {𝐴, 𝐵, 𝐶} → (𝐺 ∈ USGraph → (𝐺 ∈ FriendGraph ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸))))))
653imp1 1302 . . . . . . 7 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) → (𝐺 ∈ FriendGraph ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)))
7 prcom 4299 . . . . . . . . . . . . . . . . . 18 {𝐶, 𝐴} = {𝐴, 𝐶}
87eleq1i 2721 . . . . . . . . . . . . . . . . 17 ({𝐶, 𝐴} ∈ 𝐸 ↔ {𝐴, 𝐶} ∈ 𝐸)
98biimpi 206 . . . . . . . . . . . . . . . 16 ({𝐶, 𝐴} ∈ 𝐸 → {𝐴, 𝐶} ∈ 𝐸)
1093ad2ant3 1104 . . . . . . . . . . . . . . 15 (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸) → {𝐴, 𝐶} ∈ 𝐸)
1110adantl 481 . . . . . . . . . . . . . 14 (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)) → {𝐴, 𝐶} ∈ 𝐸)
12 simpl11 1156 . . . . . . . . . . . . . . . . 17 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) → 𝐴𝑋)
13 simpl12 1157 . . . . . . . . . . . . . . . . 17 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) → 𝐵𝑌)
14 simp1 1081 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝐵𝐴𝐶𝐵𝐶) → 𝐴𝐵)
15143ad2ant2 1103 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → 𝐴𝐵)
1615adantr 480 . . . . . . . . . . . . . . . . 17 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) → 𝐴𝐵)
1712, 13, 163jca 1261 . . . . . . . . . . . . . . . 16 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) → (𝐴𝑋𝐵𝑌𝐴𝐵))
18 simp3 1083 . . . . . . . . . . . . . . . . 17 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → 𝑉 = {𝐴, 𝐵, 𝐶})
1918anim1i 591 . . . . . . . . . . . . . . . 16 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) → (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph))
2017, 19jca 553 . . . . . . . . . . . . . . 15 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) → ((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)))
21 simp1 1081 . . . . . . . . . . . . . . 15 (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸) → {𝐴, 𝐵} ∈ 𝐸)
222, 33vfriswmgrlem 27257 . . . . . . . . . . . . . . . 16 (((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐴, 𝐵} ∈ 𝐸 → ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸))
2322imp 444 . . . . . . . . . . . . . . 15 ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸)
2420, 21, 23syl2an 493 . . . . . . . . . . . . . 14 (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)) → ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸)
2511, 24jca 553 . . . . . . . . . . . . 13 (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)) → ({𝐴, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸))
26 simpr2 1088 . . . . . . . . . . . . . 14 (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)) → {𝐵, 𝐶} ∈ 𝐸)
27 necom 2876 . . . . . . . . . . . . . . . . . . . . 21 (𝐴𝐵𝐵𝐴)
2827biimpi 206 . . . . . . . . . . . . . . . . . . . 20 (𝐴𝐵𝐵𝐴)
29283ad2ant1 1102 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝐵𝐴𝐶𝐵𝐶) → 𝐵𝐴)
30293ad2ant2 1103 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → 𝐵𝐴)
3130adantr 480 . . . . . . . . . . . . . . . . 17 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) → 𝐵𝐴)
3213, 12, 313jca 1261 . . . . . . . . . . . . . . . 16 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) → (𝐵𝑌𝐴𝑋𝐵𝐴))
33 tpcoma 4317 . . . . . . . . . . . . . . . . . 18 {𝐴, 𝐵, 𝐶} = {𝐵, 𝐴, 𝐶}
3418, 33syl6eq 2701 . . . . . . . . . . . . . . . . 17 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → 𝑉 = {𝐵, 𝐴, 𝐶})
3534anim1i 591 . . . . . . . . . . . . . . . 16 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) → (𝑉 = {𝐵, 𝐴, 𝐶} ∧ 𝐺 ∈ USGraph))
3632, 35jca 553 . . . . . . . . . . . . . . 15 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) → ((𝐵𝑌𝐴𝑋𝐵𝐴) ∧ (𝑉 = {𝐵, 𝐴, 𝐶} ∧ 𝐺 ∈ USGraph)))
37 prcom 4299 . . . . . . . . . . . . . . . . . 18 {𝐴, 𝐵} = {𝐵, 𝐴}
3837eleq1i 2721 . . . . . . . . . . . . . . . . 17 ({𝐴, 𝐵} ∈ 𝐸 ↔ {𝐵, 𝐴} ∈ 𝐸)
3938biimpi 206 . . . . . . . . . . . . . . . 16 ({𝐴, 𝐵} ∈ 𝐸 → {𝐵, 𝐴} ∈ 𝐸)
40393ad2ant1 1102 . . . . . . . . . . . . . . 15 (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸) → {𝐵, 𝐴} ∈ 𝐸)
412, 33vfriswmgrlem 27257 . . . . . . . . . . . . . . . . 17 (((𝐵𝑌𝐴𝑋𝐵𝐴) ∧ (𝑉 = {𝐵, 𝐴, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐵, 𝐴} ∈ 𝐸 → ∃!𝑤 ∈ {𝐵, 𝐴} {𝐵, 𝑤} ∈ 𝐸))
4241imp 444 . . . . . . . . . . . . . . . 16 ((((𝐵𝑌𝐴𝑋𝐵𝐴) ∧ (𝑉 = {𝐵, 𝐴, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐵, 𝐴} ∈ 𝐸) → ∃!𝑤 ∈ {𝐵, 𝐴} {𝐵, 𝑤} ∈ 𝐸)
43 reueq1 3170 . . . . . . . . . . . . . . . . 17 ({𝐴, 𝐵} = {𝐵, 𝐴} → (∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ 𝐸 ↔ ∃!𝑤 ∈ {𝐵, 𝐴} {𝐵, 𝑤} ∈ 𝐸))
4437, 43ax-mp 5 . . . . . . . . . . . . . . . 16 (∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ 𝐸 ↔ ∃!𝑤 ∈ {𝐵, 𝐴} {𝐵, 𝑤} ∈ 𝐸)
4542, 44sylibr 224 . . . . . . . . . . . . . . 15 ((((𝐵𝑌𝐴𝑋𝐵𝐴) ∧ (𝑉 = {𝐵, 𝐴, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐵, 𝐴} ∈ 𝐸) → ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ 𝐸)
4636, 40, 45syl2an 493 . . . . . . . . . . . . . 14 (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)) → ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ 𝐸)
4726, 46jca 553 . . . . . . . . . . . . 13 (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)) → ({𝐵, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ 𝐸))
4825, 47jca 553 . . . . . . . . . . . 12 (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)) → (({𝐴, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸) ∧ ({𝐵, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ 𝐸)))
49 preq1 4300 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝐴 → {𝑣, 𝐶} = {𝐴, 𝐶})
5049eleq1d 2715 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝐴 → ({𝑣, 𝐶} ∈ 𝐸 ↔ {𝐴, 𝐶} ∈ 𝐸))
51 preq1 4300 . . . . . . . . . . . . . . . . . . . 20 (𝑣 = 𝐴 → {𝑣, 𝑤} = {𝐴, 𝑤})
5251eleq1d 2715 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝐴 → ({𝑣, 𝑤} ∈ 𝐸 ↔ {𝐴, 𝑤} ∈ 𝐸))
5352reubidv 3156 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝐴 → (∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ 𝐸 ↔ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸))
5450, 53anbi12d 747 . . . . . . . . . . . . . . . . 17 (𝑣 = 𝐴 → (({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ 𝐸) ↔ ({𝐴, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸)))
55 preq1 4300 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝐵 → {𝑣, 𝐶} = {𝐵, 𝐶})
5655eleq1d 2715 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝐵 → ({𝑣, 𝐶} ∈ 𝐸 ↔ {𝐵, 𝐶} ∈ 𝐸))
57 preq1 4300 . . . . . . . . . . . . . . . . . . . 20 (𝑣 = 𝐵 → {𝑣, 𝑤} = {𝐵, 𝑤})
5857eleq1d 2715 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝐵 → ({𝑣, 𝑤} ∈ 𝐸 ↔ {𝐵, 𝑤} ∈ 𝐸))
5958reubidv 3156 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝐵 → (∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ 𝐸 ↔ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ 𝐸))
6056, 59anbi12d 747 . . . . . . . . . . . . . . . . 17 (𝑣 = 𝐵 → (({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ 𝐸) ↔ ({𝐵, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ 𝐸)))
6154, 60ralprg 4266 . . . . . . . . . . . . . . . 16 ((𝐴𝑋𝐵𝑌) → (∀𝑣 ∈ {𝐴, 𝐵} ({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ 𝐸) ↔ (({𝐴, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸) ∧ ({𝐵, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ 𝐸))))
62613adant3 1101 . . . . . . . . . . . . . . 15 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (∀𝑣 ∈ {𝐴, 𝐵} ({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ 𝐸) ↔ (({𝐴, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸) ∧ ({𝐵, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ 𝐸))))
63623ad2ant1 1102 . . . . . . . . . . . . . 14 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → (∀𝑣 ∈ {𝐴, 𝐵} ({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ 𝐸) ↔ (({𝐴, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸) ∧ ({𝐵, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ 𝐸))))
6463adantr 480 . . . . . . . . . . . . 13 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) → (∀𝑣 ∈ {𝐴, 𝐵} ({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ 𝐸) ↔ (({𝐴, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸) ∧ ({𝐵, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ 𝐸))))
6564adantr 480 . . . . . . . . . . . 12 (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)) → (∀𝑣 ∈ {𝐴, 𝐵} ({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ 𝐸) ↔ (({𝐴, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸) ∧ ({𝐵, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ 𝐸))))
6648, 65mpbird 247 . . . . . . . . . . 11 (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)) → ∀𝑣 ∈ {𝐴, 𝐵} ({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ 𝐸))
67 diftpsn3 4364 . . . . . . . . . . . . . . . 16 ((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵})
68673adant1 1099 . . . . . . . . . . . . . . 15 ((𝐴𝐵𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵})
69 reueq1 3170 . . . . . . . . . . . . . . . . 17 (({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵} → (∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ 𝐸 ↔ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ 𝐸))
7068, 69syl 17 . . . . . . . . . . . . . . . 16 ((𝐴𝐵𝐴𝐶𝐵𝐶) → (∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ 𝐸 ↔ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ 𝐸))
7170anbi2d 740 . . . . . . . . . . . . . . 15 ((𝐴𝐵𝐴𝐶𝐵𝐶) → (({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ 𝐸) ↔ ({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ 𝐸)))
7268, 71raleqbidv 3182 . . . . . . . . . . . . . 14 ((𝐴𝐵𝐴𝐶𝐵𝐶) → (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ 𝐸) ↔ ∀𝑣 ∈ {𝐴, 𝐵} ({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ 𝐸)))
73723ad2ant2 1103 . . . . . . . . . . . . 13 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ 𝐸) ↔ ∀𝑣 ∈ {𝐴, 𝐵} ({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ 𝐸)))
7473adantr 480 . . . . . . . . . . . 12 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) → (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ 𝐸) ↔ ∀𝑣 ∈ {𝐴, 𝐵} ({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ 𝐸)))
7574adantr 480 . . . . . . . . . . 11 (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)) → (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ 𝐸) ↔ ∀𝑣 ∈ {𝐴, 𝐵} ({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ 𝐸)))
7666, 75mpbird 247 . . . . . . . . . 10 (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)) → ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ 𝐸))
77763mix3d 1258 . . . . . . . . 9 (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)) → (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴})({𝑣, 𝐴} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸) ∨ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵})({𝑣, 𝐵} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵}){𝑣, 𝑤} ∈ 𝐸) ∨ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ 𝐸)))
78 sneq 4220 . . . . . . . . . . . . . . 15 ( = 𝐴 → {} = {𝐴})
7978difeq2d 3761 . . . . . . . . . . . . . 14 ( = 𝐴 → ({𝐴, 𝐵, 𝐶} ∖ {}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐴}))
80 preq2 4301 . . . . . . . . . . . . . . . 16 ( = 𝐴 → {𝑣, } = {𝑣, 𝐴})
8180eleq1d 2715 . . . . . . . . . . . . . . 15 ( = 𝐴 → ({𝑣, } ∈ 𝐸 ↔ {𝑣, 𝐴} ∈ 𝐸))
82 reueq1 3170 . . . . . . . . . . . . . . . 16 (({𝐴, 𝐵, 𝐶} ∖ {}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐴}) → (∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ 𝐸 ↔ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸))
8379, 82syl 17 . . . . . . . . . . . . . . 15 ( = 𝐴 → (∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ 𝐸 ↔ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸))
8481, 83anbi12d 747 . . . . . . . . . . . . . 14 ( = 𝐴 → (({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ 𝐸) ↔ ({𝑣, 𝐴} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸)))
8579, 84raleqbidv 3182 . . . . . . . . . . . . 13 ( = 𝐴 → (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ 𝐸) ↔ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴})({𝑣, 𝐴} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸)))
86 sneq 4220 . . . . . . . . . . . . . . 15 ( = 𝐵 → {} = {𝐵})
8786difeq2d 3761 . . . . . . . . . . . . . 14 ( = 𝐵 → ({𝐴, 𝐵, 𝐶} ∖ {}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐵}))
88 preq2 4301 . . . . . . . . . . . . . . . 16 ( = 𝐵 → {𝑣, } = {𝑣, 𝐵})
8988eleq1d 2715 . . . . . . . . . . . . . . 15 ( = 𝐵 → ({𝑣, } ∈ 𝐸 ↔ {𝑣, 𝐵} ∈ 𝐸))
90 reueq1 3170 . . . . . . . . . . . . . . . 16 (({𝐴, 𝐵, 𝐶} ∖ {}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐵}) → (∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ 𝐸 ↔ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵}){𝑣, 𝑤} ∈ 𝐸))
9187, 90syl 17 . . . . . . . . . . . . . . 15 ( = 𝐵 → (∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ 𝐸 ↔ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵}){𝑣, 𝑤} ∈ 𝐸))
9289, 91anbi12d 747 . . . . . . . . . . . . . 14 ( = 𝐵 → (({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ 𝐸) ↔ ({𝑣, 𝐵} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵}){𝑣, 𝑤} ∈ 𝐸)))
9387, 92raleqbidv 3182 . . . . . . . . . . . . 13 ( = 𝐵 → (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ 𝐸) ↔ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵})({𝑣, 𝐵} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵}){𝑣, 𝑤} ∈ 𝐸)))
94 sneq 4220 . . . . . . . . . . . . . . 15 ( = 𝐶 → {} = {𝐶})
9594difeq2d 3761 . . . . . . . . . . . . . 14 ( = 𝐶 → ({𝐴, 𝐵, 𝐶} ∖ {}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐶}))
96 preq2 4301 . . . . . . . . . . . . . . . 16 ( = 𝐶 → {𝑣, } = {𝑣, 𝐶})
9796eleq1d 2715 . . . . . . . . . . . . . . 15 ( = 𝐶 → ({𝑣, } ∈ 𝐸 ↔ {𝑣, 𝐶} ∈ 𝐸))
98 reueq1 3170 . . . . . . . . . . . . . . . 16 (({𝐴, 𝐵, 𝐶} ∖ {}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) → (∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ 𝐸 ↔ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ 𝐸))
9995, 98syl 17 . . . . . . . . . . . . . . 15 ( = 𝐶 → (∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ 𝐸 ↔ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ 𝐸))
10097, 99anbi12d 747 . . . . . . . . . . . . . 14 ( = 𝐶 → (({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ 𝐸) ↔ ({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ 𝐸)))
10195, 100raleqbidv 3182 . . . . . . . . . . . . 13 ( = 𝐶 → (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ 𝐸) ↔ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ 𝐸)))
10285, 93, 101rextpg 4269 . . . . . . . . . . . 12 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (∃ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ 𝐸) ↔ (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴})({𝑣, 𝐴} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸) ∨ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵})({𝑣, 𝐵} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵}){𝑣, 𝑤} ∈ 𝐸) ∨ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ 𝐸))))
1031023ad2ant1 1102 . . . . . . . . . . 11 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → (∃ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ 𝐸) ↔ (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴})({𝑣, 𝐴} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸) ∨ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵})({𝑣, 𝐵} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵}){𝑣, 𝑤} ∈ 𝐸) ∨ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ 𝐸))))
104103adantr 480 . . . . . . . . . 10 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) → (∃ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ 𝐸) ↔ (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴})({𝑣, 𝐴} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸) ∨ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵})({𝑣, 𝐵} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵}){𝑣, 𝑤} ∈ 𝐸) ∨ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ 𝐸))))
105104adantr 480 . . . . . . . . 9 (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)) → (∃ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ 𝐸) ↔ (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴})({𝑣, 𝐴} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸) ∨ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵})({𝑣, 𝐵} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵}){𝑣, 𝑤} ∈ 𝐸) ∨ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ 𝐸))))
10677, 105mpbird 247 . . . . . . . 8 (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)) → ∃ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ 𝐸))
107106ex 449 . . . . . . 7 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) → (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸) → ∃ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ 𝐸)))
1086, 107sylbid 230 . . . . . 6 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) → (𝐺 ∈ FriendGraph → ∃ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ 𝐸)))
109108expcom 450 . . . . 5 (𝐺 ∈ USGraph → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → (𝐺 ∈ FriendGraph → ∃ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ 𝐸))))
110109com23 86 . . . 4 (𝐺 ∈ USGraph → (𝐺 ∈ FriendGraph → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → ∃ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ 𝐸))))
1111, 110mpcom 38 . . 3 (𝐺 ∈ FriendGraph → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → ∃ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ 𝐸)))
112111com12 32 . 2 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → (𝐺 ∈ FriendGraph → ∃ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ 𝐸)))
113 difeq1 3754 . . . . . 6 (𝑉 = {𝐴, 𝐵, 𝐶} → (𝑉 ∖ {}) = ({𝐴, 𝐵, 𝐶} ∖ {}))
114 reueq1 3170 . . . . . . . 8 ((𝑉 ∖ {}) = ({𝐴, 𝐵, 𝐶} ∖ {}) → (∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸 ↔ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ 𝐸))
115113, 114syl 17 . . . . . . 7 (𝑉 = {𝐴, 𝐵, 𝐶} → (∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸 ↔ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ 𝐸))
116115anbi2d 740 . . . . . 6 (𝑉 = {𝐴, 𝐵, 𝐶} → (({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸) ↔ ({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ 𝐸)))
117113, 116raleqbidv 3182 . . . . 5 (𝑉 = {𝐴, 𝐵, 𝐶} → (∀𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸) ↔ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ 𝐸)))
118117rexeqbi1dv 3177 . . . 4 (𝑉 = {𝐴, 𝐵, 𝐶} → (∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸) ↔ ∃ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ 𝐸)))
119118imbi2d 329 . . 3 (𝑉 = {𝐴, 𝐵, 𝐶} → ((𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸)) ↔ (𝐺 ∈ FriendGraph → ∃ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ 𝐸))))
1201193ad2ant3 1104 . 2 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → ((𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸)) ↔ (𝐺 ∈ FriendGraph → ∃ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ 𝐸))))
121112, 120mpbird 247 1 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3o 1053  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  ∃!wreu 2943  cdif 3604  {csn 4210  {cpr 4212  {ctp 4214  cfv 5926  Vtxcvtx 25919  Edgcedg 25984  USGraphcusgr 26089   FriendGraph cfrgr 27236
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-hash 13158  df-edg 25985  df-umgr 26023  df-usgr 26091  df-frgr 27237
This theorem is referenced by:  1to3vfriswmgr  27260
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