Step | Hyp | Ref
| Expression |
1 | | nbgr2vtx1edg.v |
. . . . 5
Vtx   |
2 | | fvex 5889 |
. . . . 5
Vtx   |
3 | 1, 2 | eqeltri 2545 |
. . . 4
 |
4 | | hash2prb 12674 |
. . . 4
     



       |
5 | 3, 4 | ax-mp 5 |
. . 3
    



      |
6 | | simpr 468 |
. . . . . . . . . . . 12
  UHGraph

 

   |
7 | 6 | ancomd 458 |
. . . . . . . . . . 11
  UHGraph

 

   |
8 | 7 | ad2antrr 740 |
. . . . . . . . . 10
    UHGraph 
  
         
   |
9 | | id 22 |
. . . . . . . . . . . . 13
   |
10 | 9 | necomd 2698 |
. . . . . . . . . . . 12
   |
11 | 10 | adantr 472 |
. . . . . . . . . . 11
 
      |
12 | 11 | ad2antlr 741 |
. . . . . . . . . 10
    UHGraph 
  
           |
13 | | prcom 4041 |
. . . . . . . . . . . . . 14
       |
14 | 13 | eleq1i 2540 |
. . . . . . . . . . . . 13
   
     |
15 | 14 | biimpi 199 |
. . . . . . . . . . . 12
         |
16 | | sseq2 3440 |
. . . . . . . . . . . . 13
       
         |
17 | 16 | adantl 473 |
. . . . . . . . . . . 12
    
          
      |
18 | 13 | eqimssi 3472 |
. . . . . . . . . . . . 13
       |
19 | 18 | a1i 11 |
. . . . . . . . . . . 12
            |
20 | 15, 17, 19 | rspcedvd 3143 |
. . . . . . . . . . 11
    
     |
21 | 20 | adantl 473 |
. . . . . . . . . 10
    UHGraph 
  
         
     |
22 | | nbgr2vtx1edg.e |
. . . . . . . . . . . 12
Edg   |
23 | 1, 22 | nbgrel 39574 |
. . . . . . . . . . 11
 UHGraph   NeighbVtx
  

        |
24 | 23 | ad3antrrr 744 |
. . . . . . . . . 10
    UHGraph 
  
           NeighbVtx 
 

   
    |
25 | 8, 12, 21, 24 | mpbir3and 1213 |
. . . . . . . . 9
    UHGraph 
  
          NeighbVtx
   |
26 | 6 | ad2antrr 740 |
. . . . . . . . . 10
    UHGraph 
  
         
   |
27 | | simplrl 778 |
. . . . . . . . . 10
    UHGraph 
  
           |
28 | | id 22 |
. . . . . . . . . . . 12
         |
29 | | sseq2 3440 |
. . . . . . . . . . . . 13
       
         |
30 | 29 | adantl 473 |
. . . . . . . . . . . 12
    
          
      |
31 | | prcom 4041 |
. . . . . . . . . . . . . 14
       |
32 | 31 | eqimssi 3472 |
. . . . . . . . . . . . 13
       |
33 | 32 | a1i 11 |
. . . . . . . . . . . 12
            |
34 | 28, 30, 33 | rspcedvd 3143 |
. . . . . . . . . . 11
    
     |
35 | 34 | adantl 473 |
. . . . . . . . . 10
    UHGraph 
  
         
     |
36 | 1, 22 | nbgrel 39574 |
. . . . . . . . . . 11
 UHGraph   NeighbVtx
  

        |
37 | 36 | ad3antrrr 744 |
. . . . . . . . . 10
    UHGraph 
  
           NeighbVtx 
 

   
    |
38 | 26, 27, 35, 37 | mpbir3and 1213 |
. . . . . . . . 9
    UHGraph 
  
          NeighbVtx
   |
39 | 25, 38 | jca 541 |
. . . . . . . 8
    UHGraph 
  
           NeighbVtx   NeighbVtx
    |
40 | 39 | ex 441 |
. . . . . . 7
   UHGraph 
 

           NeighbVtx   NeighbVtx
     |
41 | 1, 22 | nbuhgr2vtx1edgblem 39583 |
. . . . . . . . . . . 12
  UHGraph
    NeighbVtx
 
     |
42 | 41 | 3exp 1230 |
. . . . . . . . . . 11
 UHGraph    
  NeighbVtx
        |
43 | 42 | adantr 472 |
. . . . . . . . . 10
  UHGraph

 
   
  NeighbVtx
        |
44 | 43 | adantld 474 |
. . . . . . . . 9
  UHGraph

 
 
      NeighbVtx
        |
45 | 44 | imp 436 |
. . . . . . . 8
   UHGraph 
 

       NeighbVtx        |
46 | 45 | adantld 474 |
. . . . . . 7
   UHGraph 
 

        NeighbVtx

 NeighbVtx         |
47 | 40, 46 | impbid 195 |
. . . . . 6
   UHGraph 
 

           NeighbVtx   NeighbVtx
     |
48 | | eleq1 2537 |
. . . . . . . . 9
    
      |
49 | 48 | adantl 473 |
. . . . . . . 8
 
           |
50 | | id 22 |
. . . . . . . . . 10
         |
51 | | difeq1 3533 |
. . . . . . . . . . 11
                 |
52 | 51 | raleqdv 2979 |
. . . . . . . . . 10
     
      NeighbVtx 
          NeighbVtx     |
53 | 50, 52 | raleqbidv 2987 |
. . . . . . . . 9
     

      NeighbVtx 
               NeighbVtx     |
54 | | vex 3034 |
. . . . . . . . . . 11
 |
55 | | vex 3034 |
. . . . . . . . . . 11
 |
56 | | sneq 3969 |
. . . . . . . . . . . . 13
       |
57 | 56 | difeq2d 3540 |
. . . . . . . . . . . 12
                 |
58 | | oveq2 6316 |
. . . . . . . . . . . . 13
  NeighbVtx   NeighbVtx    |
59 | 58 | eleq2d 2534 |
. . . . . . . . . . . 12
   NeighbVtx 

NeighbVtx     |
60 | 57, 59 | raleqbidv 2987 |
. . . . . . . . . . 11
  
         NeighbVtx
           NeighbVtx     |
61 | | sneq 3969 |
. . . . . . . . . . . . 13
       |
62 | 61 | difeq2d 3540 |
. . . . . . . . . . . 12
                 |
63 | | oveq2 6316 |
. . . . . . . . . . . . 13
  NeighbVtx   NeighbVtx    |
64 | 63 | eleq2d 2534 |
. . . . . . . . . . . 12
   NeighbVtx 

NeighbVtx     |
65 | 62, 64 | raleqbidv 2987 |
. . . . . . . . . . 11
  
         NeighbVtx
           NeighbVtx     |
66 | 54, 55, 60, 65 | ralpr 4016 |
. . . . . . . . . 10
 
    
         NeighbVtx 
 
         NeighbVtx  
         NeighbVtx     |
67 | | difprsn1 4099 |
. . . . . . . . . . . . 13
            |
68 | 67 | raleqdv 2979 |
. . . . . . . . . . . 12
  
         NeighbVtx
      NeighbVtx     |
69 | | eleq1 2537 |
. . . . . . . . . . . . 13
   NeighbVtx 

NeighbVtx     |
70 | 55, 69 | ralsn 4001 |
. . . . . . . . . . . 12
 
  
 NeighbVtx 

NeighbVtx    |
71 | 68, 70 | syl6bb 269 |
. . . . . . . . . . 11
  
         NeighbVtx
  NeighbVtx     |
72 | | difprsn2 4100 |
. . . . . . . . . . . . 13
            |
73 | 72 | raleqdv 2979 |
. . . . . . . . . . . 12
  
         NeighbVtx
      NeighbVtx     |
74 | | eleq1 2537 |
. . . . . . . . . . . . 13
   NeighbVtx 

NeighbVtx     |
75 | 54, 74 | ralsn 4001 |
. . . . . . . . . . . 12
 
  
 NeighbVtx 

NeighbVtx    |
76 | 73, 75 | syl6bb 269 |
. . . . . . . . . . 11
  
         NeighbVtx
  NeighbVtx     |
77 | 71, 76 | anbi12d 725 |
. . . . . . . . . 10
             NeighbVtx            NeighbVtx     NeighbVtx

 NeighbVtx      |
78 | 66, 77 | syl5bb 265 |
. . . . . . . . 9
  
    
         NeighbVtx 
  NeighbVtx

 NeighbVtx      |
79 | 53, 78 | sylan9bbr 715 |
. . . . . . . 8
 
     
       NeighbVtx    NeighbVtx   NeighbVtx
     |
80 | 49, 79 | bibi12d 328 |
. . . . . . 7
 
     

       NeighbVtx         NeighbVtx   NeighbVtx
      |
81 | 80 | adantl 473 |
. . . . . 6
   UHGraph 
 

        
      NeighbVtx  
   
  NeighbVtx

 NeighbVtx       |
82 | 47, 81 | mpbird 240 |
. . . . 5
   UHGraph 
 

       
      NeighbVtx     |
83 | 82 | ex 441 |
. . . 4
  UHGraph

 
 
      
      NeighbVtx      |
84 | 83 | rexlimdvva 2878 |
. . 3
 UHGraph  


    

       NeighbVtx      |
85 | 5, 84 | syl5bi 225 |
. 2
 UHGraph        
      NeighbVtx      |
86 | 85 | imp 436 |
1
  UHGraph
     

       NeighbVtx     |