Step |
Hyp |
Ref |
Expression |
1 |
|
erclwwlkn.w |
|- W = ( N ClWWalksN G ) |
2 |
|
erclwwlkn.r |
|- .~ = { <. t , u >. | ( t e. W /\ u e. W /\ E. n e. ( 0 ... N ) t = ( u cyclShift n ) ) } |
3 |
|
vex |
|- x e. _V |
4 |
|
vex |
|- y e. _V |
5 |
|
vex |
|- z e. _V |
6 |
1 2
|
erclwwlkneqlen |
|- ( ( x e. _V /\ y e. _V ) -> ( x .~ y -> ( # ` x ) = ( # ` y ) ) ) |
7 |
6
|
3adant3 |
|- ( ( x e. _V /\ y e. _V /\ z e. _V ) -> ( x .~ y -> ( # ` x ) = ( # ` y ) ) ) |
8 |
1 2
|
erclwwlkneqlen |
|- ( ( y e. _V /\ z e. _V ) -> ( y .~ z -> ( # ` y ) = ( # ` z ) ) ) |
9 |
8
|
3adant1 |
|- ( ( x e. _V /\ y e. _V /\ z e. _V ) -> ( y .~ z -> ( # ` y ) = ( # ` z ) ) ) |
10 |
1 2
|
erclwwlkneq |
|- ( ( y e. _V /\ z e. _V ) -> ( y .~ z <-> ( y e. W /\ z e. W /\ E. n e. ( 0 ... N ) y = ( z cyclShift n ) ) ) ) |
11 |
10
|
3adant1 |
|- ( ( x e. _V /\ y e. _V /\ z e. _V ) -> ( y .~ z <-> ( y e. W /\ z e. W /\ E. n e. ( 0 ... N ) y = ( z cyclShift n ) ) ) ) |
12 |
1 2
|
erclwwlkneq |
|- ( ( x e. _V /\ y e. _V ) -> ( x .~ y <-> ( x e. W /\ y e. W /\ E. n e. ( 0 ... N ) x = ( y cyclShift n ) ) ) ) |
13 |
12
|
3adant3 |
|- ( ( x e. _V /\ y e. _V /\ z e. _V ) -> ( x .~ y <-> ( x e. W /\ y e. W /\ E. n e. ( 0 ... N ) x = ( y cyclShift n ) ) ) ) |
14 |
|
simpr1 |
|- ( ( ( ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) /\ ( y e. W /\ z e. W /\ E. n e. ( 0 ... N ) y = ( z cyclShift n ) ) ) /\ ( x e. W /\ y e. W /\ E. n e. ( 0 ... N ) x = ( y cyclShift n ) ) ) -> x e. W ) |
15 |
|
simplr2 |
|- ( ( ( ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) /\ ( y e. W /\ z e. W /\ E. n e. ( 0 ... N ) y = ( z cyclShift n ) ) ) /\ ( x e. W /\ y e. W /\ E. n e. ( 0 ... N ) x = ( y cyclShift n ) ) ) -> z e. W ) |
16 |
|
oveq2 |
|- ( n = m -> ( y cyclShift n ) = ( y cyclShift m ) ) |
17 |
16
|
eqeq2d |
|- ( n = m -> ( x = ( y cyclShift n ) <-> x = ( y cyclShift m ) ) ) |
18 |
17
|
cbvrexvw |
|- ( E. n e. ( 0 ... N ) x = ( y cyclShift n ) <-> E. m e. ( 0 ... N ) x = ( y cyclShift m ) ) |
19 |
|
oveq2 |
|- ( n = k -> ( z cyclShift n ) = ( z cyclShift k ) ) |
20 |
19
|
eqeq2d |
|- ( n = k -> ( y = ( z cyclShift n ) <-> y = ( z cyclShift k ) ) ) |
21 |
20
|
cbvrexvw |
|- ( E. n e. ( 0 ... N ) y = ( z cyclShift n ) <-> E. k e. ( 0 ... N ) y = ( z cyclShift k ) ) |
22 |
|
eqid |
|- ( Vtx ` G ) = ( Vtx ` G ) |
23 |
22
|
clwwlknbp |
|- ( z e. ( N ClWWalksN G ) -> ( z e. Word ( Vtx ` G ) /\ ( # ` z ) = N ) ) |
24 |
|
eqcom |
|- ( ( # ` z ) = N <-> N = ( # ` z ) ) |
25 |
24
|
biimpi |
|- ( ( # ` z ) = N -> N = ( # ` z ) ) |
26 |
23 25
|
simpl2im |
|- ( z e. ( N ClWWalksN G ) -> N = ( # ` z ) ) |
27 |
26 1
|
eleq2s |
|- ( z e. W -> N = ( # ` z ) ) |
28 |
27
|
ad2antlr |
|- ( ( ( ( x e. W /\ y e. W ) /\ z e. W ) /\ ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) ) -> N = ( # ` z ) ) |
29 |
23
|
simpld |
|- ( z e. ( N ClWWalksN G ) -> z e. Word ( Vtx ` G ) ) |
30 |
29 1
|
eleq2s |
|- ( z e. W -> z e. Word ( Vtx ` G ) ) |
31 |
30
|
ad2antlr |
|- ( ( ( ( x e. W /\ y e. W ) /\ z e. W ) /\ ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) ) -> z e. Word ( Vtx ` G ) ) |
32 |
31
|
adantl |
|- ( ( N = ( # ` z ) /\ ( ( ( x e. W /\ y e. W ) /\ z e. W ) /\ ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) ) ) -> z e. Word ( Vtx ` G ) ) |
33 |
|
simprr |
|- ( ( N = ( # ` z ) /\ ( ( ( x e. W /\ y e. W ) /\ z e. W ) /\ ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) ) ) -> ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) ) |
34 |
32 33
|
cshwcsh2id |
|- ( ( N = ( # ` z ) /\ ( ( ( x e. W /\ y e. W ) /\ z e. W ) /\ ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) ) ) -> ( ( ( m e. ( 0 ... ( # ` y ) ) /\ x = ( y cyclShift m ) ) /\ ( k e. ( 0 ... ( # ` z ) ) /\ y = ( z cyclShift k ) ) ) -> E. n e. ( 0 ... ( # ` z ) ) x = ( z cyclShift n ) ) ) |
35 |
|
oveq2 |
|- ( N = ( # ` z ) -> ( 0 ... N ) = ( 0 ... ( # ` z ) ) ) |
36 |
|
oveq2 |
|- ( ( # ` z ) = ( # ` y ) -> ( 0 ... ( # ` z ) ) = ( 0 ... ( # ` y ) ) ) |
37 |
36
|
eqcoms |
|- ( ( # ` y ) = ( # ` z ) -> ( 0 ... ( # ` z ) ) = ( 0 ... ( # ` y ) ) ) |
38 |
37
|
adantr |
|- ( ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) -> ( 0 ... ( # ` z ) ) = ( 0 ... ( # ` y ) ) ) |
39 |
38
|
adantl |
|- ( ( ( ( x e. W /\ y e. W ) /\ z e. W ) /\ ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) ) -> ( 0 ... ( # ` z ) ) = ( 0 ... ( # ` y ) ) ) |
40 |
35 39
|
sylan9eq |
|- ( ( N = ( # ` z ) /\ ( ( ( x e. W /\ y e. W ) /\ z e. W ) /\ ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) ) ) -> ( 0 ... N ) = ( 0 ... ( # ` y ) ) ) |
41 |
40
|
eleq2d |
|- ( ( N = ( # ` z ) /\ ( ( ( x e. W /\ y e. W ) /\ z e. W ) /\ ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) ) ) -> ( m e. ( 0 ... N ) <-> m e. ( 0 ... ( # ` y ) ) ) ) |
42 |
41
|
anbi1d |
|- ( ( N = ( # ` z ) /\ ( ( ( x e. W /\ y e. W ) /\ z e. W ) /\ ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) ) ) -> ( ( m e. ( 0 ... N ) /\ x = ( y cyclShift m ) ) <-> ( m e. ( 0 ... ( # ` y ) ) /\ x = ( y cyclShift m ) ) ) ) |
43 |
35
|
eleq2d |
|- ( N = ( # ` z ) -> ( k e. ( 0 ... N ) <-> k e. ( 0 ... ( # ` z ) ) ) ) |
44 |
43
|
anbi1d |
|- ( N = ( # ` z ) -> ( ( k e. ( 0 ... N ) /\ y = ( z cyclShift k ) ) <-> ( k e. ( 0 ... ( # ` z ) ) /\ y = ( z cyclShift k ) ) ) ) |
45 |
44
|
adantr |
|- ( ( N = ( # ` z ) /\ ( ( ( x e. W /\ y e. W ) /\ z e. W ) /\ ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) ) ) -> ( ( k e. ( 0 ... N ) /\ y = ( z cyclShift k ) ) <-> ( k e. ( 0 ... ( # ` z ) ) /\ y = ( z cyclShift k ) ) ) ) |
46 |
42 45
|
anbi12d |
|- ( ( N = ( # ` z ) /\ ( ( ( x e. W /\ y e. W ) /\ z e. W ) /\ ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) ) ) -> ( ( ( m e. ( 0 ... N ) /\ x = ( y cyclShift m ) ) /\ ( k e. ( 0 ... N ) /\ y = ( z cyclShift k ) ) ) <-> ( ( m e. ( 0 ... ( # ` y ) ) /\ x = ( y cyclShift m ) ) /\ ( k e. ( 0 ... ( # ` z ) ) /\ y = ( z cyclShift k ) ) ) ) ) |
47 |
35
|
rexeqdv |
|- ( N = ( # ` z ) -> ( E. n e. ( 0 ... N ) x = ( z cyclShift n ) <-> E. n e. ( 0 ... ( # ` z ) ) x = ( z cyclShift n ) ) ) |
48 |
47
|
adantr |
|- ( ( N = ( # ` z ) /\ ( ( ( x e. W /\ y e. W ) /\ z e. W ) /\ ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) ) ) -> ( E. n e. ( 0 ... N ) x = ( z cyclShift n ) <-> E. n e. ( 0 ... ( # ` z ) ) x = ( z cyclShift n ) ) ) |
49 |
34 46 48
|
3imtr4d |
|- ( ( N = ( # ` z ) /\ ( ( ( x e. W /\ y e. W ) /\ z e. W ) /\ ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) ) ) -> ( ( ( m e. ( 0 ... N ) /\ x = ( y cyclShift m ) ) /\ ( k e. ( 0 ... N ) /\ y = ( z cyclShift k ) ) ) -> E. n e. ( 0 ... N ) x = ( z cyclShift n ) ) ) |
50 |
28 49
|
mpancom |
|- ( ( ( ( x e. W /\ y e. W ) /\ z e. W ) /\ ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) ) -> ( ( ( m e. ( 0 ... N ) /\ x = ( y cyclShift m ) ) /\ ( k e. ( 0 ... N ) /\ y = ( z cyclShift k ) ) ) -> E. n e. ( 0 ... N ) x = ( z cyclShift n ) ) ) |
51 |
50
|
exp5l |
|- ( ( ( ( x e. W /\ y e. W ) /\ z e. W ) /\ ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) ) -> ( m e. ( 0 ... N ) -> ( x = ( y cyclShift m ) -> ( k e. ( 0 ... N ) -> ( y = ( z cyclShift k ) -> E. n e. ( 0 ... N ) x = ( z cyclShift n ) ) ) ) ) ) |
52 |
51
|
imp41 |
|- ( ( ( ( ( ( ( x e. W /\ y e. W ) /\ z e. W ) /\ ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) ) /\ m e. ( 0 ... N ) ) /\ x = ( y cyclShift m ) ) /\ k e. ( 0 ... N ) ) -> ( y = ( z cyclShift k ) -> E. n e. ( 0 ... N ) x = ( z cyclShift n ) ) ) |
53 |
52
|
rexlimdva |
|- ( ( ( ( ( ( x e. W /\ y e. W ) /\ z e. W ) /\ ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) ) /\ m e. ( 0 ... N ) ) /\ x = ( y cyclShift m ) ) -> ( E. k e. ( 0 ... N ) y = ( z cyclShift k ) -> E. n e. ( 0 ... N ) x = ( z cyclShift n ) ) ) |
54 |
53
|
ex |
|- ( ( ( ( ( x e. W /\ y e. W ) /\ z e. W ) /\ ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) ) /\ m e. ( 0 ... N ) ) -> ( x = ( y cyclShift m ) -> ( E. k e. ( 0 ... N ) y = ( z cyclShift k ) -> E. n e. ( 0 ... N ) x = ( z cyclShift n ) ) ) ) |
55 |
54
|
rexlimdva |
|- ( ( ( ( x e. W /\ y e. W ) /\ z e. W ) /\ ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) ) -> ( E. m e. ( 0 ... N ) x = ( y cyclShift m ) -> ( E. k e. ( 0 ... N ) y = ( z cyclShift k ) -> E. n e. ( 0 ... N ) x = ( z cyclShift n ) ) ) ) |
56 |
21 55
|
syl7bi |
|- ( ( ( ( x e. W /\ y e. W ) /\ z e. W ) /\ ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) ) -> ( E. m e. ( 0 ... N ) x = ( y cyclShift m ) -> ( E. n e. ( 0 ... N ) y = ( z cyclShift n ) -> E. n e. ( 0 ... N ) x = ( z cyclShift n ) ) ) ) |
57 |
18 56
|
syl5bi |
|- ( ( ( ( x e. W /\ y e. W ) /\ z e. W ) /\ ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) ) -> ( E. n e. ( 0 ... N ) x = ( y cyclShift n ) -> ( E. n e. ( 0 ... N ) y = ( z cyclShift n ) -> E. n e. ( 0 ... N ) x = ( z cyclShift n ) ) ) ) |
58 |
57
|
exp31 |
|- ( ( x e. W /\ y e. W ) -> ( z e. W -> ( ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) -> ( E. n e. ( 0 ... N ) x = ( y cyclShift n ) -> ( E. n e. ( 0 ... N ) y = ( z cyclShift n ) -> E. n e. ( 0 ... N ) x = ( z cyclShift n ) ) ) ) ) ) |
59 |
58
|
com15 |
|- ( E. n e. ( 0 ... N ) y = ( z cyclShift n ) -> ( z e. W -> ( ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) -> ( E. n e. ( 0 ... N ) x = ( y cyclShift n ) -> ( ( x e. W /\ y e. W ) -> E. n e. ( 0 ... N ) x = ( z cyclShift n ) ) ) ) ) ) |
60 |
59
|
impcom |
|- ( ( z e. W /\ E. n e. ( 0 ... N ) y = ( z cyclShift n ) ) -> ( ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) -> ( E. n e. ( 0 ... N ) x = ( y cyclShift n ) -> ( ( x e. W /\ y e. W ) -> E. n e. ( 0 ... N ) x = ( z cyclShift n ) ) ) ) ) |
61 |
60
|
3adant1 |
|- ( ( y e. W /\ z e. W /\ E. n e. ( 0 ... N ) y = ( z cyclShift n ) ) -> ( ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) -> ( E. n e. ( 0 ... N ) x = ( y cyclShift n ) -> ( ( x e. W /\ y e. W ) -> E. n e. ( 0 ... N ) x = ( z cyclShift n ) ) ) ) ) |
62 |
61
|
impcom |
|- ( ( ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) /\ ( y e. W /\ z e. W /\ E. n e. ( 0 ... N ) y = ( z cyclShift n ) ) ) -> ( E. n e. ( 0 ... N ) x = ( y cyclShift n ) -> ( ( x e. W /\ y e. W ) -> E. n e. ( 0 ... N ) x = ( z cyclShift n ) ) ) ) |
63 |
62
|
com13 |
|- ( ( x e. W /\ y e. W ) -> ( E. n e. ( 0 ... N ) x = ( y cyclShift n ) -> ( ( ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) /\ ( y e. W /\ z e. W /\ E. n e. ( 0 ... N ) y = ( z cyclShift n ) ) ) -> E. n e. ( 0 ... N ) x = ( z cyclShift n ) ) ) ) |
64 |
63
|
3impia |
|- ( ( x e. W /\ y e. W /\ E. n e. ( 0 ... N ) x = ( y cyclShift n ) ) -> ( ( ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) /\ ( y e. W /\ z e. W /\ E. n e. ( 0 ... N ) y = ( z cyclShift n ) ) ) -> E. n e. ( 0 ... N ) x = ( z cyclShift n ) ) ) |
65 |
64
|
impcom |
|- ( ( ( ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) /\ ( y e. W /\ z e. W /\ E. n e. ( 0 ... N ) y = ( z cyclShift n ) ) ) /\ ( x e. W /\ y e. W /\ E. n e. ( 0 ... N ) x = ( y cyclShift n ) ) ) -> E. n e. ( 0 ... N ) x = ( z cyclShift n ) ) |
66 |
14 15 65
|
3jca |
|- ( ( ( ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) /\ ( y e. W /\ z e. W /\ E. n e. ( 0 ... N ) y = ( z cyclShift n ) ) ) /\ ( x e. W /\ y e. W /\ E. n e. ( 0 ... N ) x = ( y cyclShift n ) ) ) -> ( x e. W /\ z e. W /\ E. n e. ( 0 ... N ) x = ( z cyclShift n ) ) ) |
67 |
1 2
|
erclwwlkneq |
|- ( ( x e. _V /\ z e. _V ) -> ( x .~ z <-> ( x e. W /\ z e. W /\ E. n e. ( 0 ... N ) x = ( z cyclShift n ) ) ) ) |
68 |
67
|
3adant2 |
|- ( ( x e. _V /\ y e. _V /\ z e. _V ) -> ( x .~ z <-> ( x e. W /\ z e. W /\ E. n e. ( 0 ... N ) x = ( z cyclShift n ) ) ) ) |
69 |
66 68
|
syl5ibrcom |
|- ( ( ( ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) /\ ( y e. W /\ z e. W /\ E. n e. ( 0 ... N ) y = ( z cyclShift n ) ) ) /\ ( x e. W /\ y e. W /\ E. n e. ( 0 ... N ) x = ( y cyclShift n ) ) ) -> ( ( x e. _V /\ y e. _V /\ z e. _V ) -> x .~ z ) ) |
70 |
69
|
exp31 |
|- ( ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) -> ( ( y e. W /\ z e. W /\ E. n e. ( 0 ... N ) y = ( z cyclShift n ) ) -> ( ( x e. W /\ y e. W /\ E. n e. ( 0 ... N ) x = ( y cyclShift n ) ) -> ( ( x e. _V /\ y e. _V /\ z e. _V ) -> x .~ z ) ) ) ) |
71 |
70
|
com24 |
|- ( ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) -> ( ( x e. _V /\ y e. _V /\ z e. _V ) -> ( ( x e. W /\ y e. W /\ E. n e. ( 0 ... N ) x = ( y cyclShift n ) ) -> ( ( y e. W /\ z e. W /\ E. n e. ( 0 ... N ) y = ( z cyclShift n ) ) -> x .~ z ) ) ) ) |
72 |
71
|
ex |
|- ( ( # ` y ) = ( # ` z ) -> ( ( # ` x ) = ( # ` y ) -> ( ( x e. _V /\ y e. _V /\ z e. _V ) -> ( ( x e. W /\ y e. W /\ E. n e. ( 0 ... N ) x = ( y cyclShift n ) ) -> ( ( y e. W /\ z e. W /\ E. n e. ( 0 ... N ) y = ( z cyclShift n ) ) -> x .~ z ) ) ) ) ) |
73 |
72
|
com4t |
|- ( ( x e. _V /\ y e. _V /\ z e. _V ) -> ( ( x e. W /\ y e. W /\ E. n e. ( 0 ... N ) x = ( y cyclShift n ) ) -> ( ( # ` y ) = ( # ` z ) -> ( ( # ` x ) = ( # ` y ) -> ( ( y e. W /\ z e. W /\ E. n e. ( 0 ... N ) y = ( z cyclShift n ) ) -> x .~ z ) ) ) ) ) |
74 |
13 73
|
sylbid |
|- ( ( x e. _V /\ y e. _V /\ z e. _V ) -> ( x .~ y -> ( ( # ` y ) = ( # ` z ) -> ( ( # ` x ) = ( # ` y ) -> ( ( y e. W /\ z e. W /\ E. n e. ( 0 ... N ) y = ( z cyclShift n ) ) -> x .~ z ) ) ) ) ) |
75 |
74
|
com25 |
|- ( ( x e. _V /\ y e. _V /\ z e. _V ) -> ( ( y e. W /\ z e. W /\ E. n e. ( 0 ... N ) y = ( z cyclShift n ) ) -> ( ( # ` y ) = ( # ` z ) -> ( ( # ` x ) = ( # ` y ) -> ( x .~ y -> x .~ z ) ) ) ) ) |
76 |
11 75
|
sylbid |
|- ( ( x e. _V /\ y e. _V /\ z e. _V ) -> ( y .~ z -> ( ( # ` y ) = ( # ` z ) -> ( ( # ` x ) = ( # ` y ) -> ( x .~ y -> x .~ z ) ) ) ) ) |
77 |
9 76
|
mpdd |
|- ( ( x e. _V /\ y e. _V /\ z e. _V ) -> ( y .~ z -> ( ( # ` x ) = ( # ` y ) -> ( x .~ y -> x .~ z ) ) ) ) |
78 |
77
|
com24 |
|- ( ( x e. _V /\ y e. _V /\ z e. _V ) -> ( x .~ y -> ( ( # ` x ) = ( # ` y ) -> ( y .~ z -> x .~ z ) ) ) ) |
79 |
7 78
|
mpdd |
|- ( ( x e. _V /\ y e. _V /\ z e. _V ) -> ( x .~ y -> ( y .~ z -> x .~ z ) ) ) |
80 |
79
|
impd |
|- ( ( x e. _V /\ y e. _V /\ z e. _V ) -> ( ( x .~ y /\ y .~ z ) -> x .~ z ) ) |
81 |
3 4 5 80
|
mp3an |
|- ( ( x .~ y /\ y .~ z ) -> x .~ z ) |