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Theorem 3vfriswmgrlem 27039
Description: Lemma for 3vfriswmgr 27040. (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
3vfriswmgrlem (((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) → ({𝐴, 𝐵} ∈ 𝐸 → ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸))
Distinct variable groups:   𝑤,𝐴   𝑤,𝐵   𝑤,𝐶   𝑤,𝐸   𝑤,𝐺   𝑤,𝑉   𝑤,𝑋   𝑤,𝑌

Proof of Theorem 3vfriswmgrlem
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 animorr 506 . . . . . 6 ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → ({𝐴, 𝐴} ∈ 𝐸 ∨ {𝐴, 𝐵} ∈ 𝐸))
2 preq2 4246 . . . . . . . . . 10 (𝑤 = 𝐴 → {𝐴, 𝑤} = {𝐴, 𝐴})
32eleq1d 2683 . . . . . . . . 9 (𝑤 = 𝐴 → ({𝐴, 𝑤} ∈ 𝐸 ↔ {𝐴, 𝐴} ∈ 𝐸))
4 preq2 4246 . . . . . . . . . 10 (𝑤 = 𝐵 → {𝐴, 𝑤} = {𝐴, 𝐵})
54eleq1d 2683 . . . . . . . . 9 (𝑤 = 𝐵 → ({𝐴, 𝑤} ∈ 𝐸 ↔ {𝐴, 𝐵} ∈ 𝐸))
63, 5rexprg 4213 . . . . . . . 8 ((𝐴𝑋𝐵𝑌) → (∃𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸 ↔ ({𝐴, 𝐴} ∈ 𝐸 ∨ {𝐴, 𝐵} ∈ 𝐸)))
763adant3 1079 . . . . . . 7 ((𝐴𝑋𝐵𝑌𝐴𝐵) → (∃𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸 ↔ ({𝐴, 𝐴} ∈ 𝐸 ∨ {𝐴, 𝐵} ∈ 𝐸)))
87ad2antrr 761 . . . . . 6 ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → (∃𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸 ↔ ({𝐴, 𝐴} ∈ 𝐸 ∨ {𝐴, 𝐵} ∈ 𝐸)))
91, 8mpbird 247 . . . . 5 ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → ∃𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸)
10 df-rex 2914 . . . . 5 (∃𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸 ↔ ∃𝑤(𝑤 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑤} ∈ 𝐸))
119, 10sylib 208 . . . 4 ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → ∃𝑤(𝑤 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑤} ∈ 𝐸))
12 vex 3193 . . . . . . . . 9 𝑤 ∈ V
1312elpr 4176 . . . . . . . 8 (𝑤 ∈ {𝐴, 𝐵} ↔ (𝑤 = 𝐴𝑤 = 𝐵))
14 vex 3193 . . . . . . . . . . . 12 𝑦 ∈ V
1514elpr 4176 . . . . . . . . . . 11 (𝑦 ∈ {𝐴, 𝐵} ↔ (𝑦 = 𝐴𝑦 = 𝐵))
16 eqidd 2622 . . . . . . . . . . . . . . . . . 18 ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐴 = 𝐴)
1716a1i 11 . . . . . . . . . . . . . . . . 17 ({𝐴, 𝐴} ∈ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐴 = 𝐴))
1817a1i13 27 . . . . . . . . . . . . . . . 16 (𝑦 = 𝐴 → ({𝐴, 𝐴} ∈ 𝐸 → ({𝐴, 𝐴} ∈ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐴 = 𝐴))))
19 preq2 4246 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝐴 → {𝐴, 𝑦} = {𝐴, 𝐴})
2019eleq1d 2683 . . . . . . . . . . . . . . . 16 (𝑦 = 𝐴 → ({𝐴, 𝑦} ∈ 𝐸 ↔ {𝐴, 𝐴} ∈ 𝐸))
21 eqeq2 2632 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝐴 → (𝐴 = 𝑦𝐴 = 𝐴))
2221imbi2d 330 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝐴 → (((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐴 = 𝑦) ↔ ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐴 = 𝐴)))
2322imbi2d 330 . . . . . . . . . . . . . . . 16 (𝑦 = 𝐴 → (({𝐴, 𝐴} ∈ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐴 = 𝑦)) ↔ ({𝐴, 𝐴} ∈ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐴 = 𝐴))))
2418, 20, 233imtr4d 283 . . . . . . . . . . . . . . 15 (𝑦 = 𝐴 → ({𝐴, 𝑦} ∈ 𝐸 → ({𝐴, 𝐴} ∈ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐴 = 𝑦))))
25 3vfriswmgr.e . . . . . . . . . . . . . . . . . . . . . . 23 𝐸 = (Edg‘𝐺)
2625usgredgne 26025 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐺 ∈ USGraph ∧ {𝐴, 𝐴} ∈ 𝐸) → 𝐴𝐴)
2726adantll 749 . . . . . . . . . . . . . . . . . . . . 21 (((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph ) ∧ {𝐴, 𝐴} ∈ 𝐸) → 𝐴𝐴)
28 df-ne 2791 . . . . . . . . . . . . . . . . . . . . . 22 (𝐴𝐴 ↔ ¬ 𝐴 = 𝐴)
29 eqid 2621 . . . . . . . . . . . . . . . . . . . . . . 23 𝐴 = 𝐴
3029pm2.24i 146 . . . . . . . . . . . . . . . . . . . . . 22 𝐴 = 𝐴𝐴 = 𝐵)
3128, 30sylbi 207 . . . . . . . . . . . . . . . . . . . . 21 (𝐴𝐴𝐴 = 𝐵)
3227, 31syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph ) ∧ {𝐴, 𝐴} ∈ 𝐸) → 𝐴 = 𝐵)
3332ex 450 . . . . . . . . . . . . . . . . . . 19 ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph ) → ({𝐴, 𝐴} ∈ 𝐸𝐴 = 𝐵))
3433ad2antlr 762 . . . . . . . . . . . . . . . . . 18 ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → ({𝐴, 𝐴} ∈ 𝐸𝐴 = 𝐵))
3534com12 32 . . . . . . . . . . . . . . . . 17 ({𝐴, 𝐴} ∈ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐴 = 𝐵))
36352a1i 12 . . . . . . . . . . . . . . . 16 (𝑦 = 𝐵 → ({𝐴, 𝐵} ∈ 𝐸 → ({𝐴, 𝐴} ∈ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐴 = 𝐵))))
37 preq2 4246 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝐵 → {𝐴, 𝑦} = {𝐴, 𝐵})
3837eleq1d 2683 . . . . . . . . . . . . . . . 16 (𝑦 = 𝐵 → ({𝐴, 𝑦} ∈ 𝐸 ↔ {𝐴, 𝐵} ∈ 𝐸))
39 eqeq2 2632 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝐵 → (𝐴 = 𝑦𝐴 = 𝐵))
4039imbi2d 330 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝐵 → (((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐴 = 𝑦) ↔ ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐴 = 𝐵)))
4140imbi2d 330 . . . . . . . . . . . . . . . 16 (𝑦 = 𝐵 → (({𝐴, 𝐴} ∈ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐴 = 𝑦)) ↔ ({𝐴, 𝐴} ∈ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐴 = 𝐵))))
4236, 38, 413imtr4d 283 . . . . . . . . . . . . . . 15 (𝑦 = 𝐵 → ({𝐴, 𝑦} ∈ 𝐸 → ({𝐴, 𝐴} ∈ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐴 = 𝑦))))
4324, 42jaoi 394 . . . . . . . . . . . . . 14 ((𝑦 = 𝐴𝑦 = 𝐵) → ({𝐴, 𝑦} ∈ 𝐸 → ({𝐴, 𝐴} ∈ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐴 = 𝑦))))
44 eqeq1 2625 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝐴 → (𝑤 = 𝑦𝐴 = 𝑦))
4544imbi2d 330 . . . . . . . . . . . . . . . 16 (𝑤 = 𝐴 → (((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝑤 = 𝑦) ↔ ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐴 = 𝑦)))
463, 45imbi12d 334 . . . . . . . . . . . . . . 15 (𝑤 = 𝐴 → (({𝐴, 𝑤} ∈ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝑤 = 𝑦)) ↔ ({𝐴, 𝐴} ∈ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐴 = 𝑦))))
4746imbi2d 330 . . . . . . . . . . . . . 14 (𝑤 = 𝐴 → (({𝐴, 𝑦} ∈ 𝐸 → ({𝐴, 𝑤} ∈ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝑤 = 𝑦))) ↔ ({𝐴, 𝑦} ∈ 𝐸 → ({𝐴, 𝐴} ∈ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐴 = 𝑦)))))
4843, 47syl5ibr 236 . . . . . . . . . . . . 13 (𝑤 = 𝐴 → ((𝑦 = 𝐴𝑦 = 𝐵) → ({𝐴, 𝑦} ∈ 𝐸 → ({𝐴, 𝑤} ∈ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝑤 = 𝑦)))))
4929pm2.24i 146 . . . . . . . . . . . . . . . . . . . . . 22 𝐴 = 𝐴𝐵 = 𝐴)
5028, 49sylbi 207 . . . . . . . . . . . . . . . . . . . . 21 (𝐴𝐴𝐵 = 𝐴)
5127, 50syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph ) ∧ {𝐴, 𝐴} ∈ 𝐸) → 𝐵 = 𝐴)
5251ex 450 . . . . . . . . . . . . . . . . . . 19 ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph ) → ({𝐴, 𝐴} ∈ 𝐸𝐵 = 𝐴))
5352ad2antlr 762 . . . . . . . . . . . . . . . . . 18 ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → ({𝐴, 𝐴} ∈ 𝐸𝐵 = 𝐴))
5453com12 32 . . . . . . . . . . . . . . . . 17 ({𝐴, 𝐴} ∈ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐵 = 𝐴))
5554a1i13 27 . . . . . . . . . . . . . . . 16 (𝑦 = 𝐴 → ({𝐴, 𝐴} ∈ 𝐸 → ({𝐴, 𝐵} ∈ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐵 = 𝐴))))
56 eqeq2 2632 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝐴 → (𝐵 = 𝑦𝐵 = 𝐴))
5756imbi2d 330 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝐴 → (((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐵 = 𝑦) ↔ ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐵 = 𝐴)))
5857imbi2d 330 . . . . . . . . . . . . . . . 16 (𝑦 = 𝐴 → (({𝐴, 𝐵} ∈ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐵 = 𝑦)) ↔ ({𝐴, 𝐵} ∈ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐵 = 𝐴))))
5955, 20, 583imtr4d 283 . . . . . . . . . . . . . . 15 (𝑦 = 𝐴 → ({𝐴, 𝑦} ∈ 𝐸 → ({𝐴, 𝐵} ∈ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐵 = 𝑦))))
60 eqidd 2622 . . . . . . . . . . . . . . . . . 18 ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐵 = 𝐵)
6160a1i 11 . . . . . . . . . . . . . . . . 17 ({𝐴, 𝐵} ∈ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐵 = 𝐵))
6261a1i13 27 . . . . . . . . . . . . . . . 16 (𝑦 = 𝐵 → ({𝐴, 𝐵} ∈ 𝐸 → ({𝐴, 𝐵} ∈ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐵 = 𝐵))))
63 eqeq2 2632 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝐵 → (𝐵 = 𝑦𝐵 = 𝐵))
6463imbi2d 330 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝐵 → (((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐵 = 𝑦) ↔ ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐵 = 𝐵)))
6564imbi2d 330 . . . . . . . . . . . . . . . 16 (𝑦 = 𝐵 → (({𝐴, 𝐵} ∈ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐵 = 𝑦)) ↔ ({𝐴, 𝐵} ∈ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐵 = 𝐵))))
6662, 38, 653imtr4d 283 . . . . . . . . . . . . . . 15 (𝑦 = 𝐵 → ({𝐴, 𝑦} ∈ 𝐸 → ({𝐴, 𝐵} ∈ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐵 = 𝑦))))
6759, 66jaoi 394 . . . . . . . . . . . . . 14 ((𝑦 = 𝐴𝑦 = 𝐵) → ({𝐴, 𝑦} ∈ 𝐸 → ({𝐴, 𝐵} ∈ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐵 = 𝑦))))
68 eqeq1 2625 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝐵 → (𝑤 = 𝑦𝐵 = 𝑦))
6968imbi2d 330 . . . . . . . . . . . . . . . 16 (𝑤 = 𝐵 → (((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝑤 = 𝑦) ↔ ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐵 = 𝑦)))
705, 69imbi12d 334 . . . . . . . . . . . . . . 15 (𝑤 = 𝐵 → (({𝐴, 𝑤} ∈ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝑤 = 𝑦)) ↔ ({𝐴, 𝐵} ∈ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐵 = 𝑦))))
7170imbi2d 330 . . . . . . . . . . . . . 14 (𝑤 = 𝐵 → (({𝐴, 𝑦} ∈ 𝐸 → ({𝐴, 𝑤} ∈ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝑤 = 𝑦))) ↔ ({𝐴, 𝑦} ∈ 𝐸 → ({𝐴, 𝐵} ∈ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐵 = 𝑦)))))
7267, 71syl5ibr 236 . . . . . . . . . . . . 13 (𝑤 = 𝐵 → ((𝑦 = 𝐴𝑦 = 𝐵) → ({𝐴, 𝑦} ∈ 𝐸 → ({𝐴, 𝑤} ∈ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝑤 = 𝑦)))))
7348, 72jaoi 394 . . . . . . . . . . . 12 ((𝑤 = 𝐴𝑤 = 𝐵) → ((𝑦 = 𝐴𝑦 = 𝐵) → ({𝐴, 𝑦} ∈ 𝐸 → ({𝐴, 𝑤} ∈ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝑤 = 𝑦)))))
7473com3l 89 . . . . . . . . . . 11 ((𝑦 = 𝐴𝑦 = 𝐵) → ({𝐴, 𝑦} ∈ 𝐸 → ((𝑤 = 𝐴𝑤 = 𝐵) → ({𝐴, 𝑤} ∈ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝑤 = 𝑦)))))
7515, 74sylbi 207 . . . . . . . . . 10 (𝑦 ∈ {𝐴, 𝐵} → ({𝐴, 𝑦} ∈ 𝐸 → ((𝑤 = 𝐴𝑤 = 𝐵) → ({𝐴, 𝑤} ∈ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝑤 = 𝑦)))))
7675imp 445 . . . . . . . . 9 ((𝑦 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑦} ∈ 𝐸) → ((𝑤 = 𝐴𝑤 = 𝐵) → ({𝐴, 𝑤} ∈ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝑤 = 𝑦))))
7776com3l 89 . . . . . . . 8 ((𝑤 = 𝐴𝑤 = 𝐵) → ({𝐴, 𝑤} ∈ 𝐸 → ((𝑦 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑦} ∈ 𝐸) → ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝑤 = 𝑦))))
7813, 77sylbi 207 . . . . . . 7 (𝑤 ∈ {𝐴, 𝐵} → ({𝐴, 𝑤} ∈ 𝐸 → ((𝑦 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑦} ∈ 𝐸) → ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝑤 = 𝑦))))
7978imp31 448 . . . . . 6 (((𝑤 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑤} ∈ 𝐸) ∧ (𝑦 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑦} ∈ 𝐸)) → ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝑤 = 𝑦))
8079com12 32 . . . . 5 ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → (((𝑤 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑤} ∈ 𝐸) ∧ (𝑦 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑦} ∈ 𝐸)) → 𝑤 = 𝑦))
8180alrimivv 1853 . . . 4 ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → ∀𝑤𝑦(((𝑤 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑤} ∈ 𝐸) ∧ (𝑦 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑦} ∈ 𝐸)) → 𝑤 = 𝑦))
82 eleq1 2686 . . . . . 6 (𝑤 = 𝑦 → (𝑤 ∈ {𝐴, 𝐵} ↔ 𝑦 ∈ {𝐴, 𝐵}))
83 preq2 4246 . . . . . . 7 (𝑤 = 𝑦 → {𝐴, 𝑤} = {𝐴, 𝑦})
8483eleq1d 2683 . . . . . 6 (𝑤 = 𝑦 → ({𝐴, 𝑤} ∈ 𝐸 ↔ {𝐴, 𝑦} ∈ 𝐸))
8582, 84anbi12d 746 . . . . 5 (𝑤 = 𝑦 → ((𝑤 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑤} ∈ 𝐸) ↔ (𝑦 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑦} ∈ 𝐸)))
8685eu4 2517 . . . 4 (∃!𝑤(𝑤 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑤} ∈ 𝐸) ↔ (∃𝑤(𝑤 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑤} ∈ 𝐸) ∧ ∀𝑤𝑦(((𝑤 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑤} ∈ 𝐸) ∧ (𝑦 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑦} ∈ 𝐸)) → 𝑤 = 𝑦)))
8711, 81, 86sylanbrc 697 . . 3 ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → ∃!𝑤(𝑤 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑤} ∈ 𝐸))
88 df-reu 2915 . . 3 (∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸 ↔ ∃!𝑤(𝑤 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑤} ∈ 𝐸))
8987, 88sylibr 224 . 2 ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐴, 𝐵} ∈ 𝐸) → ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸)
9089ex 450 1 (((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) → ({𝐴, 𝐵} ∈ 𝐸 → ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  w3a 1036  wal 1478   = wceq 1480  wex 1701  wcel 1987  ∃!weu 2469  wne 2790  wrex 2909  ∃!wreu 2910  {cpr 4157  {ctp 4159  cfv 5857  Vtxcvtx 25808  Edgcedg 25873   USGraph cusgr 25971
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 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
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 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-card 8725  df-cda 8950  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-2 11039  df-n0 11253  df-z 11338  df-uz 11648  df-fz 12285  df-hash 13074  df-edg 25874  df-umgr 25908  df-usgr 25973
This theorem is referenced by:  3vfriswmgr  27040
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