Theorem frgr2wwlkeqm 30959

Description: If there is a (simple) path of length 2 from one vertex to another vertex and a (simple) path of length 2 from the other vertex back to the first vertex in a friendship graph, then the middle vertex is the same. This is only an observation, which is not required to proof the friendship theorem. (Contributed by Alexander van der Vekens, 20-Feb-2018.) (Revised by AV, 13-May-2021.) (Proof shortened by AV, 7-Jan-2022.)

Assertion

Ref	Expression
frgr2wwlkeqm	⊢ ((𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) → ((⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴)) → 𝑄 = 𝑃))

Proof of Theorem frgr2wwlkeqm

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp3l 1220	. . . 4 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) → 𝑃 ∈ 𝑋)
2		eqid 2763	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3	2	wwlks2onv 30569	. . . 4 ⊢ ((𝑃 ∈ 𝑋 ∧ ⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)))
4	1, 3	sylan 592	. . 3 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) ∧ ⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)))
5		simp3r 1221	. . . . . . . 8 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) → 𝑄 ∈ 𝑌)
6	2	wwlks2onv 30569	. . . . . . . 8 ⊢ ((𝑄 ∈ 𝑌 ∧ ⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴)) → (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)))
7	5, 6	sylan 592	. . . . . . 7 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) ∧ ⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴)) → (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)))
8		frgrusgr 30889	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
9		usgrumgr 29789	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UMGraph)
10	8, 9	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ UMGraph)
11	10	3ad2ant1 1151	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) → 𝐺 ∈ UMGraph)
12		simpr3 1215	. . . . . . . . . . . . . 14 ⊢ ((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) → 𝐵 ∈ (Vtx‘𝐺))
13		simpl 488	. . . . . . . . . . . . . 14 ⊢ ((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) → 𝑄 ∈ (Vtx‘𝐺))
14		simpr1 1213	. . . . . . . . . . . . . 14 ⊢ ((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) → 𝐴 ∈ (Vtx‘𝐺))
15	12, 13, 14	3jca 1146	. . . . . . . . . . . . 13 ⊢ ((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) → (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)))
16	2	wwlks2onsym 30576	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ UMGraph ∧ (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺))) → (⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴) ↔ ⟨“𝐴𝑄𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
17	11, 15, 16	syl2anr 609	. . . . . . . . . . . 12 ⊢ (((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌))) → (⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴) ↔ ⟨“𝐴𝑄𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
18		simpr1 1213	. . . . . . . . . . . . . 14 ⊢ (((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌))) → 𝐺 ∈ FriendGraph )
19		3simpb 1167	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)))
20	19	ad2antlr 740	. . . . . . . . . . . . . 14 ⊢ (((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌))) → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)))
21		simpr2 1214	. . . . . . . . . . . . . 14 ⊢ (((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌))) → 𝐴 ≠ 𝐵)
22	2	frgr2wwlkeu 30955	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ 𝐴 ≠ 𝐵) → ∃!𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))
23	18, 20, 21, 22	syl3anc 1399	. . . . . . . . . . . . 13 ⊢ (((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌))) → ∃!𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))
24		s3eq2 15046	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑄 → ⟨“𝐴𝑥𝐵”⟩ = ⟨“𝐴𝑄𝐵”⟩)
25	24	eleq1d 2849	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑄 → (⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ⟨“𝐴𝑄𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
26	25	riota2 7422	. . . . . . . . . . . . . . 15 ⊢ ((𝑄 ∈ (Vtx‘𝐺) ∧ ∃!𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → (⟨“𝐴𝑄𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ (℩𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) = 𝑄))
27	26	ad4ant14 765	. . . . . . . . . . . . . 14 ⊢ ((((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌))) ∧ ∃!𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → (⟨“𝐴𝑄𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ (℩𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) = 𝑄))
28		simplr2 1235	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌))) → 𝑃 ∈ (Vtx‘𝐺))
29		s3eq2 15046	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑃 → ⟨“𝐴𝑥𝐵”⟩ = ⟨“𝐴𝑃𝐵”⟩)
30	29	eleq1d 2849	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑃 → (⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
31	30	riota2 7422	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃 ∈ (Vtx‘𝐺) ∧ ∃!𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ (℩𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) = 𝑃))
32	28, 31	sylan 592	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌))) ∧ ∃!𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ (℩𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) = 𝑃))
33		eqtr2 2784	. . . . . . . . . . . . . . . . 17 ⊢ (((℩𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) = 𝑄 ∧ (℩𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) = 𝑃) → 𝑄 = 𝑃)
34	33	expcom 419	. . . . . . . . . . . . . . . 16 ⊢ ((℩𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) = 𝑃 → ((℩𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) = 𝑄 → 𝑄 = 𝑃))
35	32, 34	biimtrdi 256	. . . . . . . . . . . . . . 15 ⊢ ((((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌))) ∧ ∃!𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → ((℩𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) = 𝑄 → 𝑄 = 𝑃)))
36	35	com23 87	. . . . . . . . . . . . . 14 ⊢ ((((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌))) ∧ ∃!𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → ((℩𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) = 𝑄 → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → 𝑄 = 𝑃)))
37	27, 36	sylbid 243	. . . . . . . . . . . . 13 ⊢ ((((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌))) ∧ ∃!𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → (⟨“𝐴𝑄𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → 𝑄 = 𝑃)))
38	23, 37	mpdan 700	. . . . . . . . . . . 12 ⊢ (((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌))) → (⟨“𝐴𝑄𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → 𝑄 = 𝑃)))
39	17, 38	sylbid 243	. . . . . . . . . . 11 ⊢ (((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌))) → (⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴) → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → 𝑄 = 𝑃)))
40	39	expimpd 459	. . . . . . . . . 10 ⊢ ((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) → (((𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) ∧ ⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴)) → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → 𝑄 = 𝑃)))
41	40	ex 418	. . . . . . . . 9 ⊢ (𝑄 ∈ (Vtx‘𝐺) → ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → (((𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) ∧ ⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴)) → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → 𝑄 = 𝑃))))
42	41	com23 87	. . . . . . . 8 ⊢ (𝑄 ∈ (Vtx‘𝐺) → (((𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) ∧ ⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴)) → ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → 𝑄 = 𝑃))))
43	42	3ad2ant2 1152	. . . . . . 7 ⊢ ((𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) → (((𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) ∧ ⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴)) → ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → 𝑄 = 𝑃))))
44	7, 43	mpcom 39	. . . . . 6 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) ∧ ⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴)) → ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → 𝑄 = 𝑃)))
45	44	ex 418	. . . . 5 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) → (⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴) → ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → 𝑄 = 𝑃))))
46	45	com24 96	. . . 4 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → (⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴) → 𝑄 = 𝑃))))
47	46	imp 412	. . 3 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) ∧ ⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → (⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴) → 𝑄 = 𝑃)))
48	4, 47	mpd 16	. 2 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) ∧ ⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → (⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴) → 𝑄 = 𝑃))
49	48	expimpd 459	1 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) → ((⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴)) → 𝑄 = 𝑃))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∃!wreu 3367  ‘cfv 6555  ℩crio 7396  (class class class)co 7440  2c2 12421  ⟨“cs3 15018  Vtxcvtx 29601  UMGraphcumgr 29686  USGraphcusgr 29757   WWalksNOn cwwlksnon 30443   FriendGraph cfrgr 30886

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-ac2 10565  ax-cnex 11280  ax-resscn 11281  ax-1cn 11282  ax-icn 11283  ax-addcl 11284  ax-addrcl 11285  ax-mulcl 11286  ax-mulrcl 11287  ax-mulcom 11288  ax-addass 11289  ax-mulass 11290  ax-distr 11291  ax-i2m1 11292  ax-1ne0 11293  ax-1rid 11294  ax-rnegex 11295  ax-rrecex 11296  ax-cnre 11297  ax-pre-lttri 11298  ax-pre-lttrn 11299  ax-pre-ltadd 11300  ax-pre-mulgt0 11301

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-isom 6564  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-1st 8022  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-1o 8501  df-2o 8502  df-oadd 8505  df-er 8742  df-map 8874  df-pm 8875  df-en 8999  df-dom 9000  df-sdom 9001  df-fin 9002  df-dju 10006  df-card 10044  df-ac 10219  df-pnf 11369  df-mnf 11370  df-xr 11371  df-ltxr 11372  df-le 11373  df-sub 11567  df-neg 11568  df-nn 12360  df-2 12429  df-3 12430  df-n0 12631  df-xnn0 12704  df-z 12718  df-uz 12990  df-fz 13664  df-fzo 13813  df-hash 14499  df-word 14683  df-concat 14740  df-s1 14767  df-s2 15024  df-s3 15025  df-edg 29653  df-uhgr 29663  df-upgr 29687  df-umgr 29688  df-uspgr 29758  df-usgr 29759  df-wlks 30207  df-wwlks 30446  df-wwlksn 30447  df-wwlksnon 30448  df-frgr 30887

This theorem is used by: (None)