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Theorem frg2woteqm 25264
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  E  /\  A  =/= 
B )  ->  (
( <. A ,  P ,  B >.  e.  ( A ( V 2WalksOnOt  E ) B )  /\  <. B ,  Q ,  A >.  e.  ( 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 25082 . . . 4  |-  ( <. A ,  P ,  B >.  e.  ( A ( V 2WalksOnOt  E ) B )  ->  (
( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  P  e.  V  /\  B  e.  V
) ) )
2 2wlkonotv 25082 . . . 4  |-  ( <. B ,  Q ,  A >.  e.  ( B ( V 2WalksOnOt  E ) A )  ->  (
( V  e.  _V  /\  E  e.  _V )  /\  ( B  e.  V  /\  Q  e.  V  /\  A  e.  V
) ) )
31, 2anim12i 564 . . 3  |-  ( (
<. A ,  P ,  B >.  e.  ( A ( V 2WalksOnOt  E ) B )  /\  <. B ,  Q ,  A >.  e.  ( B ( V 2WalksOnOt  E ) A ) )  ->  ( (
( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  P  e.  V  /\  B  e.  V
) )  /\  (
( V  e.  _V  /\  E  e.  _V )  /\  ( B  e.  V  /\  Q  e.  V  /\  A  e.  V
) ) ) )
4 frisusgra 25197 . . . . . . . . . . . . 13  |-  ( V FriendGrph  E  ->  V USGrph  E )
54adantr 463 . . . . . . . . . . . 12  |-  ( ( V FriendGrph  E  /\  A  =/= 
B )  ->  V USGrph  E )
6 simprr3 1044 . . . . . . . . . . . . 13  |-  ( ( P  e.  V  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( B  e.  V  /\  Q  e.  V  /\  A  e.  V )
) )  ->  A  e.  V )
7 simpl 455 . . . . . . . . . . . . 13  |-  ( ( P  e.  V  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( B  e.  V  /\  Q  e.  V  /\  A  e.  V )
) )  ->  P  e.  V )
8 simprr1 1042 . . . . . . . . . . . . 13  |-  ( ( P  e.  V  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( B  e.  V  /\  Q  e.  V  /\  A  e.  V )
) )  ->  B  e.  V )
96, 7, 83jca 1174 . . . . . . . . . . . 12  |-  ( ( P  e.  V  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( B  e.  V  /\  Q  e.  V  /\  A  e.  V )
) )  ->  ( A  e.  V  /\  P  e.  V  /\  B  e.  V )
)
10 usg2wlkonot 25088 . . . . . . . . . . . 12  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  P  e.  V  /\  B  e.  V )
)  ->  ( <. A ,  P ,  B >.  e.  ( A ( V 2WalksOnOt  E ) B )  <-> 
( { A ,  P }  e.  ran  E  /\  { P ,  B }  e.  ran  E ) ) )
115, 9, 10syl2anr 476 . . . . . . . . . . 11  |-  ( ( ( P  e.  V  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( B  e.  V  /\  Q  e.  V  /\  A  e.  V )
) )  /\  ( V FriendGrph  E  /\  A  =/= 
B ) )  -> 
( <. A ,  P ,  B >.  e.  ( A ( V 2WalksOnOt  E ) B )  <->  ( { A ,  P }  e.  ran  E  /\  { P ,  B }  e.  ran  E ) ) )
12 simprr 755 . . . . . . . . . . . 12  |-  ( ( P  e.  V  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( B  e.  V  /\  Q  e.  V  /\  A  e.  V )
) )  ->  ( B  e.  V  /\  Q  e.  V  /\  A  e.  V )
)
13 usg2wlkonot 25088 . . . . . . . . . . . 12  |-  ( ( V USGrph  E  /\  ( B  e.  V  /\  Q  e.  V  /\  A  e.  V )
)  ->  ( <. B ,  Q ,  A >.  e.  ( B ( V 2WalksOnOt  E ) A )  <-> 
( { B ,  Q }  e.  ran  E  /\  { Q ,  A }  e.  ran  E ) ) )
145, 12, 13syl2anr 476 . . . . . . . . . . 11  |-  ( ( ( P  e.  V  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( B  e.  V  /\  Q  e.  V  /\  A  e.  V )
) )  /\  ( V FriendGrph  E  /\  A  =/= 
B ) )  -> 
( <. B ,  Q ,  A >.  e.  ( B ( V 2WalksOnOt  E ) A )  <->  ( { B ,  Q }  e.  ran  E  /\  { Q ,  A }  e.  ran  E ) ) )
1511, 14anbi12d 708 . . . . . . . . . 10  |-  ( ( ( P  e.  V  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( B  e.  V  /\  Q  e.  V  /\  A  e.  V )
) )  /\  ( V FriendGrph  E  /\  A  =/= 
B ) )  -> 
( ( <. A ,  P ,  B >.  e.  ( A ( V 2WalksOnOt  E ) B )  /\  <. B ,  Q ,  A >.  e.  ( B ( V 2WalksOnOt  E ) A ) )  <->  ( ( { A ,  P }  e.  ran  E  /\  { P ,  B }  e.  ran  E )  /\  ( { B ,  Q }  e.  ran  E  /\  { Q ,  A }  e.  ran  E ) ) ) )
16 simpl 455 . . . . . . . . . . . . 13  |-  ( ( V FriendGrph  E  /\  A  =/= 
B )  ->  V FriendGrph  E )
1716adantl 464 . . . . . . . . . . . 12  |-  ( ( ( P  e.  V  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( B  e.  V  /\  Q  e.  V  /\  A  e.  V )
) )  /\  ( V FriendGrph  E  /\  A  =/= 
B ) )  ->  V FriendGrph  E )
188adantr 463 . . . . . . . . . . . . 13  |-  ( ( ( P  e.  V  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( B  e.  V  /\  Q  e.  V  /\  A  e.  V )
) )  /\  ( V FriendGrph  E  /\  A  =/= 
B ) )  ->  B  e.  V )
196adantr 463 . . . . . . . . . . . . 13  |-  ( ( ( P  e.  V  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( B  e.  V  /\  Q  e.  V  /\  A  e.  V )
) )  /\  ( V FriendGrph  E  /\  A  =/= 
B ) )  ->  A  e.  V )
20 necom 2723 . . . . . . . . . . . . . . 15  |-  ( A  =/=  B  <->  B  =/=  A )
2120biimpi 194 . . . . . . . . . . . . . 14  |-  ( A  =/=  B  ->  B  =/=  A )
2221ad2antll 726 . . . . . . . . . . . . 13  |-  ( ( ( P  e.  V  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( B  e.  V  /\  Q  e.  V  /\  A  e.  V )
) )  /\  ( V FriendGrph  E  /\  A  =/= 
B ) )  ->  B  =/=  A )
2318, 19, 223jca 1174 . . . . . . . . . . . 12  |-  ( ( ( P  e.  V  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( B  e.  V  /\  Q  e.  V  /\  A  e.  V )
) )  /\  ( V FriendGrph  E  /\  A  =/= 
B ) )  -> 
( B  e.  V  /\  A  e.  V  /\  B  =/=  A
) )
24 frgraeu 25259 . . . . . . . . . . . 12  |-  ( V FriendGrph  E  ->  ( ( B  e.  V  /\  A  e.  V  /\  B  =/= 
A )  ->  E! p ( { B ,  p }  e.  ran  E  /\  { p ,  A }  e.  ran  E ) ) )
2517, 23, 24sylc 60 . . . . . . . . . . 11  |-  ( ( ( P  e.  V  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( B  e.  V  /\  Q  e.  V  /\  A  e.  V )
) )  /\  ( V FriendGrph  E  /\  A  =/= 
B ) )  ->  E! p ( { B ,  p }  e.  ran  E  /\  { p ,  A }  e.  ran  E ) )
26 preq2 4096 . . . . . . . . . . . . . . 15  |-  ( p  =  q  ->  { B ,  p }  =  { B ,  q }
)
2726eleq1d 2523 . . . . . . . . . . . . . 14  |-  ( p  =  q  ->  ( { B ,  p }  e.  ran  E  <->  { B ,  q }  e.  ran  E ) )
28 preq1 4095 . . . . . . . . . . . . . . 15  |-  ( p  =  q  ->  { p ,  A }  =  {
q ,  A }
)
2928eleq1d 2523 . . . . . . . . . . . . . 14  |-  ( p  =  q  ->  ( { p ,  A }  e.  ran  E  <->  { q ,  A }  e.  ran  E ) )
3027, 29anbi12d 708 . . . . . . . . . . . . 13  |-  ( p  =  q  ->  (
( { B ,  p }  e.  ran  E  /\  { p ,  A }  e.  ran  E )  <->  ( { B ,  q }  e.  ran  E  /\  { q ,  A }  e.  ran  E ) ) )
3130eu4 2336 . . . . . . . . . . . 12  |-  ( E! p ( { B ,  p }  e.  ran  E  /\  { p ,  A }  e.  ran  E )  <->  ( E. p
( { B ,  p }  e.  ran  E  /\  { p ,  A }  e.  ran  E )  /\  A. p A. q ( ( ( { B ,  p }  e.  ran  E  /\  { p ,  A }  e.  ran  E )  /\  ( { B ,  q }  e.  ran  E  /\  { q ,  A }  e.  ran  E ) )  ->  p  =  q ) ) )
32 preq2 4096 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( p  =  P  ->  { B ,  p }  =  { B ,  P }
)
33 prcom 4094 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  { B ,  P }  =  { P ,  B }
3432, 33syl6eq 2511 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( p  =  P  ->  { B ,  p }  =  { P ,  B }
)
3534eleq1d 2523 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( p  =  P  ->  ( { B ,  p }  e.  ran  E  <->  { P ,  B }  e.  ran  E ) )
36 preq1 4095 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( p  =  P  ->  { p ,  A }  =  { P ,  A }
)
37 prcom 4094 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  { P ,  A }  =  { A ,  P }
3836, 37syl6eq 2511 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( p  =  P  ->  { p ,  A }  =  { A ,  P }
)
3938eleq1d 2523 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( p  =  P  ->  ( { p ,  A }  e.  ran  E  <->  { A ,  P }  e.  ran  E ) )
4035, 39anbi12d 708 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( p  =  P  ->  (
( { B ,  p }  e.  ran  E  /\  { p ,  A }  e.  ran  E )  <->  ( { P ,  B }  e.  ran  E  /\  { A ,  P }  e.  ran  E ) ) )
41 ancom 448 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( { P ,  B }  e.  ran  E  /\  { A ,  P }  e.  ran  E )  <->  ( { A ,  P }  e.  ran  E  /\  { P ,  B }  e.  ran  E ) )
4240, 41syl6bb 261 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( p  =  P  ->  (
( { B ,  p }  e.  ran  E  /\  { p ,  A }  e.  ran  E )  <->  ( { A ,  P }  e.  ran  E  /\  { P ,  B }  e.  ran  E ) ) )
43 preq2 4096 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( q  =  Q  ->  { B ,  q }  =  { B ,  Q }
)
4443eleq1d 2523 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( q  =  Q  ->  ( { B ,  q }  e.  ran  E  <->  { B ,  Q }  e.  ran  E ) )
45 preq1 4095 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( q  =  Q  ->  { q ,  A }  =  { Q ,  A }
)
4645eleq1d 2523 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( q  =  Q  ->  ( { q ,  A }  e.  ran  E  <->  { Q ,  A }  e.  ran  E ) )
4744, 46anbi12d 708 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( q  =  Q  ->  (
( { B , 
q }  e.  ran  E  /\  { q ,  A }  e.  ran  E )  <->  ( { B ,  Q }  e.  ran  E  /\  { Q ,  A }  e.  ran  E ) ) )
4842, 47bi2anan9 871 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( p  =  P  /\  q  =  Q )  ->  ( ( ( { B ,  p }  e.  ran  E  /\  {
p ,  A }  e.  ran  E )  /\  ( { B ,  q }  e.  ran  E  /\  { q ,  A }  e.  ran  E ) )  <->  ( ( { A ,  P }  e.  ran  E  /\  { P ,  B }  e.  ran  E )  /\  ( { B ,  Q }  e.  ran  E  /\  { Q ,  A }  e.  ran  E ) ) ) )
49 eqeq12 2473 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( p  =  P  /\  q  =  Q )  ->  ( p  =  q  <-> 
P  =  Q ) )
50 eqcom 2463 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( P  =  Q  <->  Q  =  P )
5149, 50syl6bb 261 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( p  =  P  /\  q  =  Q )  ->  ( p  =  q  <-> 
Q  =  P ) )
5248, 51imbi12d 318 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( p  =  P  /\  q  =  Q )  ->  ( ( ( ( { B ,  p }  e.  ran  E  /\  { p ,  A }  e.  ran  E )  /\  ( { B ,  q }  e.  ran  E  /\  { q ,  A }  e.  ran  E ) )  ->  p  =  q )  <->  ( (
( { A ,  P }  e.  ran  E  /\  { P ,  B }  e.  ran  E )  /\  ( { B ,  Q }  e.  ran  E  /\  { Q ,  A }  e.  ran  E ) )  ->  Q  =  P ) ) )
5352spc2gv 3194 . . . . . . . . . . . . . . . . . . 19  |-  ( ( P  e.  V  /\  Q  e.  V )  ->  ( A. p A. q ( ( ( { B ,  p }  e.  ran  E  /\  { p ,  A }  e.  ran  E )  /\  ( { B ,  q }  e.  ran  E  /\  { q ,  A }  e.  ran  E ) )  ->  p  =  q )  ->  (
( ( { A ,  P }  e.  ran  E  /\  { P ,  B }  e.  ran  E )  /\  ( { B ,  Q }  e.  ran  E  /\  { Q ,  A }  e.  ran  E ) )  ->  Q  =  P ) ) )
5453expcom 433 . . . . . . . . . . . . . . . . . 18  |-  ( Q  e.  V  ->  ( P  e.  V  ->  ( A. p A. q
( ( ( { B ,  p }  e.  ran  E  /\  {
p ,  A }  e.  ran  E )  /\  ( { B ,  q }  e.  ran  E  /\  { q ,  A }  e.  ran  E ) )  ->  p  =  q )  ->  (
( ( { A ,  P }  e.  ran  E  /\  { P ,  B }  e.  ran  E )  /\  ( { B ,  Q }  e.  ran  E  /\  { Q ,  A }  e.  ran  E ) )  ->  Q  =  P ) ) ) )
55543ad2ant2 1016 . . . . . . . . . . . . . . . . 17  |-  ( ( B  e.  V  /\  Q  e.  V  /\  A  e.  V )  ->  ( P  e.  V  ->  ( A. p A. q ( ( ( { B ,  p }  e.  ran  E  /\  { p ,  A }  e.  ran  E )  /\  ( { B ,  q }  e.  ran  E  /\  { q ,  A }  e.  ran  E ) )  ->  p  =  q )  ->  (
( ( { A ,  P }  e.  ran  E  /\  { P ,  B }  e.  ran  E )  /\  ( { B ,  Q }  e.  ran  E  /\  { Q ,  A }  e.  ran  E ) )  ->  Q  =  P ) ) ) )
5655adantl 464 . . . . . . . . . . . . . . . 16  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( B  e.  V  /\  Q  e.  V  /\  A  e.  V
) )  ->  ( P  e.  V  ->  ( A. p A. q
( ( ( { B ,  p }  e.  ran  E  /\  {
p ,  A }  e.  ran  E )  /\  ( { B ,  q }  e.  ran  E  /\  { q ,  A }  e.  ran  E ) )  ->  p  =  q )  ->  (
( ( { A ,  P }  e.  ran  E  /\  { P ,  B }  e.  ran  E )  /\  ( { B ,  Q }  e.  ran  E  /\  { Q ,  A }  e.  ran  E ) )  ->  Q  =  P ) ) ) )
5756impcom 428 . . . . . . . . . . . . . . 15  |-  ( ( P  e.  V  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( B  e.  V  /\  Q  e.  V  /\  A  e.  V )
) )  ->  ( A. p A. q ( ( ( { B ,  p }  e.  ran  E  /\  { p ,  A }  e.  ran  E )  /\  ( { B ,  q }  e.  ran  E  /\  { q ,  A }  e.  ran  E ) )  ->  p  =  q )  ->  ( (
( { A ,  P }  e.  ran  E  /\  { P ,  B }  e.  ran  E )  /\  ( { B ,  Q }  e.  ran  E  /\  { Q ,  A }  e.  ran  E ) )  ->  Q  =  P ) ) )
5857adantr 463 . . . . . . . . . . . . . 14  |-  ( ( ( P  e.  V  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( B  e.  V  /\  Q  e.  V  /\  A  e.  V )
) )  /\  ( V FriendGrph  E  /\  A  =/= 
B ) )  -> 
( A. p A. q ( ( ( { B ,  p }  e.  ran  E  /\  { p ,  A }  e.  ran  E )  /\  ( { B ,  q }  e.  ran  E  /\  { q ,  A }  e.  ran  E ) )  ->  p  =  q )  ->  (
( ( { A ,  P }  e.  ran  E  /\  { P ,  B }  e.  ran  E )  /\  ( { B ,  Q }  e.  ran  E  /\  { Q ,  A }  e.  ran  E ) )  ->  Q  =  P ) ) )
5958com12 31 . . . . . . . . . . . . 13  |-  ( A. p A. q ( ( ( { B ,  p }  e.  ran  E  /\  { p ,  A }  e.  ran  E )  /\  ( { B ,  q }  e.  ran  E  /\  { q ,  A }  e.  ran  E ) )  ->  p  =  q )  ->  ( (
( P  e.  V  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( B  e.  V  /\  Q  e.  V  /\  A  e.  V )
) )  /\  ( V FriendGrph  E  /\  A  =/= 
B ) )  -> 
( ( ( { A ,  P }  e.  ran  E  /\  { P ,  B }  e.  ran  E )  /\  ( { B ,  Q }  e.  ran  E  /\  { Q ,  A }  e.  ran  E ) )  ->  Q  =  P ) ) )
6059adantl 464 . . . . . . . . . . . 12  |-  ( ( E. p ( { B ,  p }  e.  ran  E  /\  {
p ,  A }  e.  ran  E )  /\  A. p A. q ( ( ( { B ,  p }  e.  ran  E  /\  { p ,  A }  e.  ran  E )  /\  ( { B ,  q }  e.  ran  E  /\  { q ,  A }  e.  ran  E ) )  ->  p  =  q ) )  ->  (
( ( P  e.  V  /\  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( B  e.  V  /\  Q  e.  V  /\  A  e.  V
) ) )  /\  ( V FriendGrph  E  /\  A  =/=  B ) )  -> 
( ( ( { A ,  P }  e.  ran  E  /\  { P ,  B }  e.  ran  E )  /\  ( { B ,  Q }  e.  ran  E  /\  { Q ,  A }  e.  ran  E ) )  ->  Q  =  P ) ) )
6131, 60sylbi 195 . . . . . . . . . . 11  |-  ( E! p ( { B ,  p }  e.  ran  E  /\  { p ,  A }  e.  ran  E )  ->  ( (
( P  e.  V  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( B  e.  V  /\  Q  e.  V  /\  A  e.  V )
) )  /\  ( V FriendGrph  E  /\  A  =/= 
B ) )  -> 
( ( ( { A ,  P }  e.  ran  E  /\  { P ,  B }  e.  ran  E )  /\  ( { B ,  Q }  e.  ran  E  /\  { Q ,  A }  e.  ran  E ) )  ->  Q  =  P ) ) )
6225, 61mpcom 36 . . . . . . . . . 10  |-  ( ( ( P  e.  V  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( B  e.  V  /\  Q  e.  V  /\  A  e.  V )
) )  /\  ( V FriendGrph  E  /\  A  =/= 
B ) )  -> 
( ( ( { A ,  P }  e.  ran  E  /\  { P ,  B }  e.  ran  E )  /\  ( { B ,  Q }  e.  ran  E  /\  { Q ,  A }  e.  ran  E ) )  ->  Q  =  P ) )
6315, 62sylbid 215 . . . . . . . . 9  |-  ( ( ( P  e.  V  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( B  e.  V  /\  Q  e.  V  /\  A  e.  V )
) )  /\  ( V FriendGrph  E  /\  A  =/= 
B ) )  -> 
( ( <. A ,  P ,  B >.  e.  ( A ( V 2WalksOnOt  E ) B )  /\  <. B ,  Q ,  A >.  e.  ( B ( V 2WalksOnOt  E ) A ) )  ->  Q  =  P )
)
6463ex 432 . . . . . . . 8  |-  ( ( P  e.  V  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( B  e.  V  /\  Q  e.  V  /\  A  e.  V )
) )  ->  (
( V FriendGrph  E  /\  A  =/=  B )  ->  (
( <. A ,  P ,  B >.  e.  ( A ( V 2WalksOnOt  E ) B )  /\  <. B ,  Q ,  A >.  e.  ( B ( V 2WalksOnOt  E ) A ) )  ->  Q  =  P ) ) )
6564com23 78 . . . . . . 7  |-  ( ( P  e.  V  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( B  e.  V  /\  Q  e.  V  /\  A  e.  V )
) )  ->  (
( <. A ,  P ,  B >.  e.  ( A ( V 2WalksOnOt  E ) B )  /\  <. B ,  Q ,  A >.  e.  ( B ( V 2WalksOnOt  E ) A ) )  ->  ( ( V FriendGrph  E  /\  A  =/= 
B )  ->  Q  =  P ) ) )
6665ex 432 . . . . . 6  |-  ( P  e.  V  ->  (
( ( V  e. 
_V  /\  E  e.  _V )  /\  ( B  e.  V  /\  Q  e.  V  /\  A  e.  V )
)  ->  ( ( <. A ,  P ,  B >.  e.  ( A ( V 2WalksOnOt  E ) B )  /\  <. B ,  Q ,  A >.  e.  ( B ( V 2WalksOnOt  E ) A ) )  ->  ( ( V FriendGrph  E  /\  A  =/= 
B )  ->  Q  =  P ) ) ) )
67663ad2ant2 1016 . . . . 5  |-  ( ( A  e.  V  /\  P  e.  V  /\  B  e.  V )  ->  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( B  e.  V  /\  Q  e.  V  /\  A  e.  V )
)  ->  ( ( <. A ,  P ,  B >.  e.  ( A ( V 2WalksOnOt  E ) B )  /\  <. B ,  Q ,  A >.  e.  ( B ( V 2WalksOnOt  E ) A ) )  ->  ( ( V FriendGrph  E  /\  A  =/= 
B )  ->  Q  =  P ) ) ) )
6867adantl 464 . . . 4  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  P  e.  V  /\  B  e.  V
) )  ->  (
( ( V  e. 
_V  /\  E  e.  _V )  /\  ( B  e.  V  /\  Q  e.  V  /\  A  e.  V )
)  ->  ( ( <. A ,  P ,  B >.  e.  ( A ( V 2WalksOnOt  E ) B )  /\  <. B ,  Q ,  A >.  e.  ( B ( V 2WalksOnOt  E ) A ) )  ->  ( ( V FriendGrph  E  /\  A  =/= 
B )  ->  Q  =  P ) ) ) )
6968imp 427 . . 3  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( A  e.  V  /\  P  e.  V  /\  B  e.  V )
)  /\  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( B  e.  V  /\  Q  e.  V  /\  A  e.  V )
) )  ->  (
( <. A ,  P ,  B >.  e.  ( A ( V 2WalksOnOt  E ) B )  /\  <. B ,  Q ,  A >.  e.  ( B ( V 2WalksOnOt  E ) A ) )  ->  ( ( V FriendGrph  E  /\  A  =/= 
B )  ->  Q  =  P ) ) )
703, 69mpcom 36 . 2  |-  ( (
<. A ,  P ,  B >.  e.  ( A ( V 2WalksOnOt  E ) B )  /\  <. B ,  Q ,  A >.  e.  ( B ( V 2WalksOnOt  E ) A ) )  ->  ( ( V FriendGrph  E  /\  A  =/= 
B )  ->  Q  =  P ) )
7170com12 31 1  |-  ( ( V FriendGrph  E  /\  A  =/= 
B )  ->  (
( <. A ,  P ,  B >.  e.  ( A ( V 2WalksOnOt  E ) B )  /\  <. B ,  Q ,  A >.  e.  ( B ( V 2WalksOnOt  E ) A ) )  ->  Q  =  P ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971   A.wal 1396    = wceq 1398   E.wex 1617    e. wcel 1823   E!weu 2284    =/= wne 2649   _Vcvv 3106   {cpr 4018   <.cotp 4024   class class class wbr 4439   ran crn 4989  (class class class)co 6270   USGrph cusg 24535   2WalksOnOt c2wlkonot 25060   FriendGrph cfrgra 25193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-ot 4025  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-hash 12391  df-word 12529  df-usgra 24538  df-wlk 24713  df-wlkon 24719  df-2wlkonot 25063  df-frgra 25194
This theorem is referenced by:  frg2woteq  25265
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