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Theorem frg2woteqm 26586
 Description: There is a (simple) path of length 2 from one vertex to another vertex in a friendship graph if and only if there is a (simple) path of length 2 from the other vertex back to the first vertex. (Contributed by Alexander van der Vekens, 20-Feb-2018.)
Assertion
Ref Expression
frg2woteqm ((𝑉 FriendGrph 𝐸𝐴𝐵) → ((⟨𝐴, 𝑃, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ⟨𝐵, 𝑄, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴)) → 𝑄 = 𝑃))

Proof of Theorem frg2woteqm
Dummy variables 𝑝 𝑞 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 2wlkonotv 26404 . . . 4 (⟨𝐴, 𝑃, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝑃𝑉𝐵𝑉)))
2 2wlkonotv 26404 . . . 4 (⟨𝐵, 𝑄, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐵𝑉𝑄𝑉𝐴𝑉)))
31, 2anim12i 588 . . 3 ((⟨𝐴, 𝑃, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ⟨𝐵, 𝑄, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴)) → (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝑃𝑉𝐵𝑉)) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐵𝑉𝑄𝑉𝐴𝑉))))
4 frisusgra 26519 . . . . . . . . . . . . 13 (𝑉 FriendGrph 𝐸𝑉 USGrph 𝐸)
54adantr 480 . . . . . . . . . . . 12 ((𝑉 FriendGrph 𝐸𝐴𝐵) → 𝑉 USGrph 𝐸)
6 simprr3 1104 . . . . . . . . . . . . 13 ((𝑃𝑉 ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐵𝑉𝑄𝑉𝐴𝑉))) → 𝐴𝑉)
7 simpl 472 . . . . . . . . . . . . 13 ((𝑃𝑉 ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐵𝑉𝑄𝑉𝐴𝑉))) → 𝑃𝑉)
8 simprr1 1102 . . . . . . . . . . . . 13 ((𝑃𝑉 ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐵𝑉𝑄𝑉𝐴𝑉))) → 𝐵𝑉)
96, 7, 83jca 1235 . . . . . . . . . . . 12 ((𝑃𝑉 ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐵𝑉𝑄𝑉𝐴𝑉))) → (𝐴𝑉𝑃𝑉𝐵𝑉))
10 usg2wlkonot 26410 . . . . . . . . . . . 12 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝑃𝑉𝐵𝑉)) → (⟨𝐴, 𝑃, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ↔ ({𝐴, 𝑃} ∈ ran 𝐸 ∧ {𝑃, 𝐵} ∈ ran 𝐸)))
115, 9, 10syl2anr 494 . . . . . . . . . . 11 (((𝑃𝑉 ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐵𝑉𝑄𝑉𝐴𝑉))) ∧ (𝑉 FriendGrph 𝐸𝐴𝐵)) → (⟨𝐴, 𝑃, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ↔ ({𝐴, 𝑃} ∈ ran 𝐸 ∧ {𝑃, 𝐵} ∈ ran 𝐸)))
12 simprr 792 . . . . . . . . . . . 12 ((𝑃𝑉 ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐵𝑉𝑄𝑉𝐴𝑉))) → (𝐵𝑉𝑄𝑉𝐴𝑉))
13 usg2wlkonot 26410 . . . . . . . . . . . 12 ((𝑉 USGrph 𝐸 ∧ (𝐵𝑉𝑄𝑉𝐴𝑉)) → (⟨𝐵, 𝑄, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴) ↔ ({𝐵, 𝑄} ∈ ran 𝐸 ∧ {𝑄, 𝐴} ∈ ran 𝐸)))
145, 12, 13syl2anr 494 . . . . . . . . . . 11 (((𝑃𝑉 ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐵𝑉𝑄𝑉𝐴𝑉))) ∧ (𝑉 FriendGrph 𝐸𝐴𝐵)) → (⟨𝐵, 𝑄, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴) ↔ ({𝐵, 𝑄} ∈ ran 𝐸 ∧ {𝑄, 𝐴} ∈ ran 𝐸)))
1511, 14anbi12d 743 . . . . . . . . . 10 (((𝑃𝑉 ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐵𝑉𝑄𝑉𝐴𝑉))) ∧ (𝑉 FriendGrph 𝐸𝐴𝐵)) → ((⟨𝐴, 𝑃, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ⟨𝐵, 𝑄, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴)) ↔ (({𝐴, 𝑃} ∈ ran 𝐸 ∧ {𝑃, 𝐵} ∈ ran 𝐸) ∧ ({𝐵, 𝑄} ∈ ran 𝐸 ∧ {𝑄, 𝐴} ∈ ran 𝐸))))
16 simpl 472 . . . . . . . . . . . . 13 ((𝑉 FriendGrph 𝐸𝐴𝐵) → 𝑉 FriendGrph 𝐸)
1716adantl 481 . . . . . . . . . . . 12 (((𝑃𝑉 ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐵𝑉𝑄𝑉𝐴𝑉))) ∧ (𝑉 FriendGrph 𝐸𝐴𝐵)) → 𝑉 FriendGrph 𝐸)
188adantr 480 . . . . . . . . . . . . 13 (((𝑃𝑉 ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐵𝑉𝑄𝑉𝐴𝑉))) ∧ (𝑉 FriendGrph 𝐸𝐴𝐵)) → 𝐵𝑉)
196adantr 480 . . . . . . . . . . . . 13 (((𝑃𝑉 ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐵𝑉𝑄𝑉𝐴𝑉))) ∧ (𝑉 FriendGrph 𝐸𝐴𝐵)) → 𝐴𝑉)
20 necom 2835 . . . . . . . . . . . . . . 15 (𝐴𝐵𝐵𝐴)
2120biimpi 205 . . . . . . . . . . . . . 14 (𝐴𝐵𝐵𝐴)
2221ad2antll 761 . . . . . . . . . . . . 13 (((𝑃𝑉 ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐵𝑉𝑄𝑉𝐴𝑉))) ∧ (𝑉 FriendGrph 𝐸𝐴𝐵)) → 𝐵𝐴)
2318, 19, 223jca 1235 . . . . . . . . . . . 12 (((𝑃𝑉 ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐵𝑉𝑄𝑉𝐴𝑉))) ∧ (𝑉 FriendGrph 𝐸𝐴𝐵)) → (𝐵𝑉𝐴𝑉𝐵𝐴))
24 frgraeu 26581 . . . . . . . . . . . 12 (𝑉 FriendGrph 𝐸 → ((𝐵𝑉𝐴𝑉𝐵𝐴) → ∃!𝑝({𝐵, 𝑝} ∈ ran 𝐸 ∧ {𝑝, 𝐴} ∈ ran 𝐸)))
2517, 23, 24sylc 63 . . . . . . . . . . 11 (((𝑃𝑉 ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐵𝑉𝑄𝑉𝐴𝑉))) ∧ (𝑉 FriendGrph 𝐸𝐴𝐵)) → ∃!𝑝({𝐵, 𝑝} ∈ ran 𝐸 ∧ {𝑝, 𝐴} ∈ ran 𝐸))
26 preq2 4213 . . . . . . . . . . . . . . 15 (𝑝 = 𝑞 → {𝐵, 𝑝} = {𝐵, 𝑞})
2726eleq1d 2672 . . . . . . . . . . . . . 14 (𝑝 = 𝑞 → ({𝐵, 𝑝} ∈ ran 𝐸 ↔ {𝐵, 𝑞} ∈ ran 𝐸))
28 preq1 4212 . . . . . . . . . . . . . . 15 (𝑝 = 𝑞 → {𝑝, 𝐴} = {𝑞, 𝐴})
2928eleq1d 2672 . . . . . . . . . . . . . 14 (𝑝 = 𝑞 → ({𝑝, 𝐴} ∈ ran 𝐸 ↔ {𝑞, 𝐴} ∈ ran 𝐸))
3027, 29anbi12d 743 . . . . . . . . . . . . 13 (𝑝 = 𝑞 → (({𝐵, 𝑝} ∈ ran 𝐸 ∧ {𝑝, 𝐴} ∈ ran 𝐸) ↔ ({𝐵, 𝑞} ∈ ran 𝐸 ∧ {𝑞, 𝐴} ∈ ran 𝐸)))
3130eu4 2506 . . . . . . . . . . . 12 (∃!𝑝({𝐵, 𝑝} ∈ ran 𝐸 ∧ {𝑝, 𝐴} ∈ ran 𝐸) ↔ (∃𝑝({𝐵, 𝑝} ∈ ran 𝐸 ∧ {𝑝, 𝐴} ∈ ran 𝐸) ∧ ∀𝑝𝑞((({𝐵, 𝑝} ∈ ran 𝐸 ∧ {𝑝, 𝐴} ∈ ran 𝐸) ∧ ({𝐵, 𝑞} ∈ ran 𝐸 ∧ {𝑞, 𝐴} ∈ ran 𝐸)) → 𝑝 = 𝑞)))
32 preq2 4213 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑝 = 𝑃 → {𝐵, 𝑝} = {𝐵, 𝑃})
33 prcom 4211 . . . . . . . . . . . . . . . . . . . . . . . . . 26 {𝐵, 𝑃} = {𝑃, 𝐵}
3432, 33syl6eq 2660 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑝 = 𝑃 → {𝐵, 𝑝} = {𝑃, 𝐵})
3534eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑝 = 𝑃 → ({𝐵, 𝑝} ∈ ran 𝐸 ↔ {𝑃, 𝐵} ∈ ran 𝐸))
36 preq1 4212 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑝 = 𝑃 → {𝑝, 𝐴} = {𝑃, 𝐴})
37 prcom 4211 . . . . . . . . . . . . . . . . . . . . . . . . . 26 {𝑃, 𝐴} = {𝐴, 𝑃}
3836, 37syl6eq 2660 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑝 = 𝑃 → {𝑝, 𝐴} = {𝐴, 𝑃})
3938eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑝 = 𝑃 → ({𝑝, 𝐴} ∈ ran 𝐸 ↔ {𝐴, 𝑃} ∈ ran 𝐸))
4035, 39anbi12d 743 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑝 = 𝑃 → (({𝐵, 𝑝} ∈ ran 𝐸 ∧ {𝑝, 𝐴} ∈ ran 𝐸) ↔ ({𝑃, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝑃} ∈ ran 𝐸)))
41 ancom 465 . . . . . . . . . . . . . . . . . . . . . . 23 (({𝑃, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝑃} ∈ ran 𝐸) ↔ ({𝐴, 𝑃} ∈ ran 𝐸 ∧ {𝑃, 𝐵} ∈ ran 𝐸))
4240, 41syl6bb 275 . . . . . . . . . . . . . . . . . . . . . 22 (𝑝 = 𝑃 → (({𝐵, 𝑝} ∈ ran 𝐸 ∧ {𝑝, 𝐴} ∈ ran 𝐸) ↔ ({𝐴, 𝑃} ∈ ran 𝐸 ∧ {𝑃, 𝐵} ∈ ran 𝐸)))
43 preq2 4213 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑞 = 𝑄 → {𝐵, 𝑞} = {𝐵, 𝑄})
4443eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑞 = 𝑄 → ({𝐵, 𝑞} ∈ ran 𝐸 ↔ {𝐵, 𝑄} ∈ ran 𝐸))
45 preq1 4212 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑞 = 𝑄 → {𝑞, 𝐴} = {𝑄, 𝐴})
4645eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑞 = 𝑄 → ({𝑞, 𝐴} ∈ ran 𝐸 ↔ {𝑄, 𝐴} ∈ ran 𝐸))
4744, 46anbi12d 743 . . . . . . . . . . . . . . . . . . . . . 22 (𝑞 = 𝑄 → (({𝐵, 𝑞} ∈ ran 𝐸 ∧ {𝑞, 𝐴} ∈ ran 𝐸) ↔ ({𝐵, 𝑄} ∈ ran 𝐸 ∧ {𝑄, 𝐴} ∈ ran 𝐸)))
4842, 47bi2anan9 913 . . . . . . . . . . . . . . . . . . . . 21 ((𝑝 = 𝑃𝑞 = 𝑄) → ((({𝐵, 𝑝} ∈ ran 𝐸 ∧ {𝑝, 𝐴} ∈ ran 𝐸) ∧ ({𝐵, 𝑞} ∈ ran 𝐸 ∧ {𝑞, 𝐴} ∈ ran 𝐸)) ↔ (({𝐴, 𝑃} ∈ ran 𝐸 ∧ {𝑃, 𝐵} ∈ ran 𝐸) ∧ ({𝐵, 𝑄} ∈ ran 𝐸 ∧ {𝑄, 𝐴} ∈ ran 𝐸))))
49 eqeq12 2623 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑝 = 𝑃𝑞 = 𝑄) → (𝑝 = 𝑞𝑃 = 𝑄))
50 eqcom 2617 . . . . . . . . . . . . . . . . . . . . . 22 (𝑃 = 𝑄𝑄 = 𝑃)
5149, 50syl6bb 275 . . . . . . . . . . . . . . . . . . . . 21 ((𝑝 = 𝑃𝑞 = 𝑄) → (𝑝 = 𝑞𝑄 = 𝑃))
5248, 51imbi12d 333 . . . . . . . . . . . . . . . . . . . 20 ((𝑝 = 𝑃𝑞 = 𝑄) → (((({𝐵, 𝑝} ∈ ran 𝐸 ∧ {𝑝, 𝐴} ∈ ran 𝐸) ∧ ({𝐵, 𝑞} ∈ ran 𝐸 ∧ {𝑞, 𝐴} ∈ ran 𝐸)) → 𝑝 = 𝑞) ↔ ((({𝐴, 𝑃} ∈ ran 𝐸 ∧ {𝑃, 𝐵} ∈ ran 𝐸) ∧ ({𝐵, 𝑄} ∈ ran 𝐸 ∧ {𝑄, 𝐴} ∈ ran 𝐸)) → 𝑄 = 𝑃)))
5352spc2gv 3269 . . . . . . . . . . . . . . . . . . 19 ((𝑃𝑉𝑄𝑉) → (∀𝑝𝑞((({𝐵, 𝑝} ∈ ran 𝐸 ∧ {𝑝, 𝐴} ∈ ran 𝐸) ∧ ({𝐵, 𝑞} ∈ ran 𝐸 ∧ {𝑞, 𝐴} ∈ ran 𝐸)) → 𝑝 = 𝑞) → ((({𝐴, 𝑃} ∈ ran 𝐸 ∧ {𝑃, 𝐵} ∈ ran 𝐸) ∧ ({𝐵, 𝑄} ∈ ran 𝐸 ∧ {𝑄, 𝐴} ∈ ran 𝐸)) → 𝑄 = 𝑃)))
5453expcom 450 . . . . . . . . . . . . . . . . . 18 (𝑄𝑉 → (𝑃𝑉 → (∀𝑝𝑞((({𝐵, 𝑝} ∈ ran 𝐸 ∧ {𝑝, 𝐴} ∈ ran 𝐸) ∧ ({𝐵, 𝑞} ∈ ran 𝐸 ∧ {𝑞, 𝐴} ∈ ran 𝐸)) → 𝑝 = 𝑞) → ((({𝐴, 𝑃} ∈ ran 𝐸 ∧ {𝑃, 𝐵} ∈ ran 𝐸) ∧ ({𝐵, 𝑄} ∈ ran 𝐸 ∧ {𝑄, 𝐴} ∈ ran 𝐸)) → 𝑄 = 𝑃))))
55543ad2ant2 1076 . . . . . . . . . . . . . . . . 17 ((𝐵𝑉𝑄𝑉𝐴𝑉) → (𝑃𝑉 → (∀𝑝𝑞((({𝐵, 𝑝} ∈ ran 𝐸 ∧ {𝑝, 𝐴} ∈ ran 𝐸) ∧ ({𝐵, 𝑞} ∈ ran 𝐸 ∧ {𝑞, 𝐴} ∈ ran 𝐸)) → 𝑝 = 𝑞) → ((({𝐴, 𝑃} ∈ ran 𝐸 ∧ {𝑃, 𝐵} ∈ ran 𝐸) ∧ ({𝐵, 𝑄} ∈ ran 𝐸 ∧ {𝑄, 𝐴} ∈ ran 𝐸)) → 𝑄 = 𝑃))))
5655adantl 481 . . . . . . . . . . . . . . . 16 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐵𝑉𝑄𝑉𝐴𝑉)) → (𝑃𝑉 → (∀𝑝𝑞((({𝐵, 𝑝} ∈ ran 𝐸 ∧ {𝑝, 𝐴} ∈ ran 𝐸) ∧ ({𝐵, 𝑞} ∈ ran 𝐸 ∧ {𝑞, 𝐴} ∈ ran 𝐸)) → 𝑝 = 𝑞) → ((({𝐴, 𝑃} ∈ ran 𝐸 ∧ {𝑃, 𝐵} ∈ ran 𝐸) ∧ ({𝐵, 𝑄} ∈ ran 𝐸 ∧ {𝑄, 𝐴} ∈ ran 𝐸)) → 𝑄 = 𝑃))))
5756impcom 445 . . . . . . . . . . . . . . 15 ((𝑃𝑉 ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐵𝑉𝑄𝑉𝐴𝑉))) → (∀𝑝𝑞((({𝐵, 𝑝} ∈ ran 𝐸 ∧ {𝑝, 𝐴} ∈ ran 𝐸) ∧ ({𝐵, 𝑞} ∈ ran 𝐸 ∧ {𝑞, 𝐴} ∈ ran 𝐸)) → 𝑝 = 𝑞) → ((({𝐴, 𝑃} ∈ ran 𝐸 ∧ {𝑃, 𝐵} ∈ ran 𝐸) ∧ ({𝐵, 𝑄} ∈ ran 𝐸 ∧ {𝑄, 𝐴} ∈ ran 𝐸)) → 𝑄 = 𝑃)))
5857adantr 480 . . . . . . . . . . . . . 14 (((𝑃𝑉 ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐵𝑉𝑄𝑉𝐴𝑉))) ∧ (𝑉 FriendGrph 𝐸𝐴𝐵)) → (∀𝑝𝑞((({𝐵, 𝑝} ∈ ran 𝐸 ∧ {𝑝, 𝐴} ∈ ran 𝐸) ∧ ({𝐵, 𝑞} ∈ ran 𝐸 ∧ {𝑞, 𝐴} ∈ ran 𝐸)) → 𝑝 = 𝑞) → ((({𝐴, 𝑃} ∈ ran 𝐸 ∧ {𝑃, 𝐵} ∈ ran 𝐸) ∧ ({𝐵, 𝑄} ∈ ran 𝐸 ∧ {𝑄, 𝐴} ∈ ran 𝐸)) → 𝑄 = 𝑃)))
5958com12 32 . . . . . . . . . . . . 13 (∀𝑝𝑞((({𝐵, 𝑝} ∈ ran 𝐸 ∧ {𝑝, 𝐴} ∈ ran 𝐸) ∧ ({𝐵, 𝑞} ∈ ran 𝐸 ∧ {𝑞, 𝐴} ∈ ran 𝐸)) → 𝑝 = 𝑞) → (((𝑃𝑉 ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐵𝑉𝑄𝑉𝐴𝑉))) ∧ (𝑉 FriendGrph 𝐸𝐴𝐵)) → ((({𝐴, 𝑃} ∈ ran 𝐸 ∧ {𝑃, 𝐵} ∈ ran 𝐸) ∧ ({𝐵, 𝑄} ∈ ran 𝐸 ∧ {𝑄, 𝐴} ∈ ran 𝐸)) → 𝑄 = 𝑃)))
6059adantl 481 . . . . . . . . . . . 12 ((∃𝑝({𝐵, 𝑝} ∈ ran 𝐸 ∧ {𝑝, 𝐴} ∈ ran 𝐸) ∧ ∀𝑝𝑞((({𝐵, 𝑝} ∈ ran 𝐸 ∧ {𝑝, 𝐴} ∈ ran 𝐸) ∧ ({𝐵, 𝑞} ∈ ran 𝐸 ∧ {𝑞, 𝐴} ∈ ran 𝐸)) → 𝑝 = 𝑞)) → (((𝑃𝑉 ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐵𝑉𝑄𝑉𝐴𝑉))) ∧ (𝑉 FriendGrph 𝐸𝐴𝐵)) → ((({𝐴, 𝑃} ∈ ran 𝐸 ∧ {𝑃, 𝐵} ∈ ran 𝐸) ∧ ({𝐵, 𝑄} ∈ ran 𝐸 ∧ {𝑄, 𝐴} ∈ ran 𝐸)) → 𝑄 = 𝑃)))
6131, 60sylbi 206 . . . . . . . . . . 11 (∃!𝑝({𝐵, 𝑝} ∈ ran 𝐸 ∧ {𝑝, 𝐴} ∈ ran 𝐸) → (((𝑃𝑉 ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐵𝑉𝑄𝑉𝐴𝑉))) ∧ (𝑉 FriendGrph 𝐸𝐴𝐵)) → ((({𝐴, 𝑃} ∈ ran 𝐸 ∧ {𝑃, 𝐵} ∈ ran 𝐸) ∧ ({𝐵, 𝑄} ∈ ran 𝐸 ∧ {𝑄, 𝐴} ∈ ran 𝐸)) → 𝑄 = 𝑃)))
6225, 61mpcom 37 . . . . . . . . . 10 (((𝑃𝑉 ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐵𝑉𝑄𝑉𝐴𝑉))) ∧ (𝑉 FriendGrph 𝐸𝐴𝐵)) → ((({𝐴, 𝑃} ∈ ran 𝐸 ∧ {𝑃, 𝐵} ∈ ran 𝐸) ∧ ({𝐵, 𝑄} ∈ ran 𝐸 ∧ {𝑄, 𝐴} ∈ ran 𝐸)) → 𝑄 = 𝑃))
6315, 62sylbid 229 . . . . . . . . 9 (((𝑃𝑉 ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐵𝑉𝑄𝑉𝐴𝑉))) ∧ (𝑉 FriendGrph 𝐸𝐴𝐵)) → ((⟨𝐴, 𝑃, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ⟨𝐵, 𝑄, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴)) → 𝑄 = 𝑃))
6463ex 449 . . . . . . . 8 ((𝑃𝑉 ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐵𝑉𝑄𝑉𝐴𝑉))) → ((𝑉 FriendGrph 𝐸𝐴𝐵) → ((⟨𝐴, 𝑃, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ⟨𝐵, 𝑄, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴)) → 𝑄 = 𝑃)))
6564com23 84 . . . . . . 7 ((𝑃𝑉 ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐵𝑉𝑄𝑉𝐴𝑉))) → ((⟨𝐴, 𝑃, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ⟨𝐵, 𝑄, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴)) → ((𝑉 FriendGrph 𝐸𝐴𝐵) → 𝑄 = 𝑃)))
6665ex 449 . . . . . 6 (𝑃𝑉 → (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐵𝑉𝑄𝑉𝐴𝑉)) → ((⟨𝐴, 𝑃, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ⟨𝐵, 𝑄, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴)) → ((𝑉 FriendGrph 𝐸𝐴𝐵) → 𝑄 = 𝑃))))
67663ad2ant2 1076 . . . . 5 ((𝐴𝑉𝑃𝑉𝐵𝑉) → (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐵𝑉𝑄𝑉𝐴𝑉)) → ((⟨𝐴, 𝑃, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ⟨𝐵, 𝑄, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴)) → ((𝑉 FriendGrph 𝐸𝐴𝐵) → 𝑄 = 𝑃))))
6867adantl 481 . . . 4 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝑃𝑉𝐵𝑉)) → (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐵𝑉𝑄𝑉𝐴𝑉)) → ((⟨𝐴, 𝑃, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ⟨𝐵, 𝑄, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴)) → ((𝑉 FriendGrph 𝐸𝐴𝐵) → 𝑄 = 𝑃))))
6968imp 444 . . 3 ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝑃𝑉𝐵𝑉)) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐵𝑉𝑄𝑉𝐴𝑉))) → ((⟨𝐴, 𝑃, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ⟨𝐵, 𝑄, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴)) → ((𝑉 FriendGrph 𝐸𝐴𝐵) → 𝑄 = 𝑃)))
703, 69mpcom 37 . 2 ((⟨𝐴, 𝑃, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ⟨𝐵, 𝑄, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴)) → ((𝑉 FriendGrph 𝐸𝐴𝐵) → 𝑄 = 𝑃))
7170com12 32 1 ((𝑉 FriendGrph 𝐸𝐴𝐵) → ((⟨𝐴, 𝑃, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ⟨𝐵, 𝑄, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴)) → 𝑄 = 𝑃))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃!weu 2458   ≠ wne 2780  Vcvv 3173  {cpr 4127  ⟨cotp 4133   class class class wbr 4583  ran crn 5039  (class class class)co 6549   USGrph cusg 25859   2WalksOnOt c2wlkonot 26382   FriendGrph cfrgra 26515 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-ot 4134  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-map 7746  df-pm 7747  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-fzo 12335  df-hash 12980  df-word 13154  df-usgra 25862  df-wlk 26036  df-wlkon 26042  df-2wlkonot 26385  df-frgra 26516 This theorem is referenced by:  frg2woteq  26587
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