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Theorem frg2woteqm 25866
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 25684 . . . 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 25684 . . . 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 576 . . 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 25799 . . . . . . . . . . . . 13  |-  ( V FriendGrph  E  ->  V USGrph  E )
54adantr 472 . . . . . . . . . . . 12  |-  ( ( V FriendGrph  E  /\  A  =/= 
B )  ->  V USGrph  E )
6 simprr3 1080 . . . . . . . . . . . . 13  |-  ( ( P  e.  V  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( B  e.  V  /\  Q  e.  V  /\  A  e.  V )
) )  ->  A  e.  V )
7 simpl 464 . . . . . . . . . . . . 13  |-  ( ( P  e.  V  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( B  e.  V  /\  Q  e.  V  /\  A  e.  V )
) )  ->  P  e.  V )
8 simprr1 1078 . . . . . . . . . . . . 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 1210 . . . . . . . . . . . 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 25690 . . . . . . . . . . . 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 486 . . . . . . . . . . 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 774 . . . . . . . . . . . 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 25690 . . . . . . . . . . . 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 486 . . . . . . . . . . 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 725 . . . . . . . . . 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 464 . . . . . . . . . . . . 13  |-  ( ( V FriendGrph  E  /\  A  =/= 
B )  ->  V FriendGrph  E )
1716adantl 473 . . . . . . . . . . . 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 472 . . . . . . . . . . . . 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 472 . . . . . . . . . . . . 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 2696 . . . . . . . . . . . . . . 15  |-  ( A  =/=  B  <->  B  =/=  A )
2120biimpi 199 . . . . . . . . . . . . . 14  |-  ( A  =/=  B  ->  B  =/=  A )
2221ad2antll 743 . . . . . . . . . . . . 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 1210 . . . . . . . . . . . 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 25861 . . . . . . . . . . . 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 61 . . . . . . . . . . 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 4043 . . . . . . . . . . . . . . 15  |-  ( p  =  q  ->  { B ,  p }  =  { B ,  q }
)
2726eleq1d 2533 . . . . . . . . . . . . . 14  |-  ( p  =  q  ->  ( { B ,  p }  e.  ran  E  <->  { B ,  q }  e.  ran  E ) )
28 preq1 4042 . . . . . . . . . . . . . . 15  |-  ( p  =  q  ->  { p ,  A }  =  {
q ,  A }
)
2928eleq1d 2533 . . . . . . . . . . . . . 14  |-  ( p  =  q  ->  ( { p ,  A }  e.  ran  E  <->  { q ,  A }  e.  ran  E ) )
3027, 29anbi12d 725 . . . . . . . . . . . . 13  |-  ( p  =  q  ->  (
( { B ,  p }  e.  ran  E  /\  { p ,  A }  e.  ran  E )  <->  ( { B ,  q }  e.  ran  E  /\  { q ,  A }  e.  ran  E ) ) )
3130eu4 2367 . . . . . . . . . . . 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 4043 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( p  =  P  ->  { B ,  p }  =  { B ,  P }
)
33 prcom 4041 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  { B ,  P }  =  { P ,  B }
3432, 33syl6eq 2521 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( p  =  P  ->  { B ,  p }  =  { P ,  B }
)
3534eleq1d 2533 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( p  =  P  ->  ( { B ,  p }  e.  ran  E  <->  { P ,  B }  e.  ran  E ) )
36 preq1 4042 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( p  =  P  ->  { p ,  A }  =  { P ,  A }
)
37 prcom 4041 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  { P ,  A }  =  { A ,  P }
3836, 37syl6eq 2521 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( p  =  P  ->  { p ,  A }  =  { A ,  P }
)
3938eleq1d 2533 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( p  =  P  ->  ( { p ,  A }  e.  ran  E  <->  { A ,  P }  e.  ran  E ) )
4035, 39anbi12d 725 . . . . . . . . . . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( { P ,  B }  e.  ran  E  /\  { A ,  P }  e.  ran  E )  <->  ( { A ,  P }  e.  ran  E  /\  { P ,  B }  e.  ran  E ) )
4240, 41syl6bb 269 . . . . . . . . . . . . . . . . . . . . . 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 4043 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( q  =  Q  ->  { B ,  q }  =  { B ,  Q }
)
4443eleq1d 2533 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( q  =  Q  ->  ( { B ,  q }  e.  ran  E  <->  { B ,  Q }  e.  ran  E ) )
45 preq1 4042 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( q  =  Q  ->  { q ,  A }  =  { Q ,  A }
)
4645eleq1d 2533 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( q  =  Q  ->  ( { q ,  A }  e.  ran  E  <->  { Q ,  A }  e.  ran  E ) )
4744, 46anbi12d 725 . . . . . . . . . . . . . . . . . . . . . 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 890 . . . . . . . . . . . . . . . . . . . . 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 2484 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( p  =  P  /\  q  =  Q )  ->  ( p  =  q  <-> 
P  =  Q ) )
50 eqcom 2478 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( P  =  Q  <->  Q  =  P )
5149, 50syl6bb 269 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( p  =  P  /\  q  =  Q )  ->  ( p  =  q  <-> 
Q  =  P ) )
5248, 51imbi12d 327 . . . . . . . . . . . . . . . . . . . 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 3123 . . . . . . . . . . . . . . . . . . 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 442 . . . . . . . . . . . . . . . . . 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 1052 . . . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . . 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 437 . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . 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 473 . . . . . . . . . . . 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 200 . . . . . . . . . . 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 223 . . . . . . . . 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 441 . . . . . . . 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 80 . . . . . . 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 441 . . . . . 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 1052 . . . . 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 473 . . . 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 436 . . 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 189    /\ wa 376    /\ w3a 1007   A.wal 1450    = wceq 1452   E.wex 1671    e. wcel 1904   E!weu 2319    =/= wne 2641   _Vcvv 3031   {cpr 3961   <.cotp 3967   class class class wbr 4395   ran crn 4840  (class class class)co 6308   USGrph cusg 25136   2WalksOnOt c2wlkonot 25662   FriendGrph cfrgra 25795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-ot 3968  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943  df-hash 12554  df-word 12711  df-usgra 25139  df-wlk 25315  df-wlkon 25321  df-2wlkonot 25665  df-frgra 25796
This theorem is referenced by:  frg2woteq  25867
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