Step |
Hyp |
Ref |
Expression |
1 |
|
opex |
|- <. <. x , y >. , z >. e. _V |
2 |
|
opex |
|- <. x , y >. e. _V |
3 |
|
vex |
|- z e. _V |
4 |
2 3
|
eqvinop |
|- ( w = <. <. x , y >. , z >. <-> E. a E. t ( w = <. a , t >. /\ <. a , t >. = <. <. x , y >. , z >. ) ) |
5 |
4
|
biimpi |
|- ( w = <. <. x , y >. , z >. -> E. a E. t ( w = <. a , t >. /\ <. a , t >. = <. <. x , y >. , z >. ) ) |
6 |
|
eqeq1 |
|- ( w = <. a , t >. -> ( w = <. <. x , y >. , z >. <-> <. a , t >. = <. <. x , y >. , z >. ) ) |
7 |
|
vex |
|- a e. _V |
8 |
|
vex |
|- t e. _V |
9 |
7 8
|
opth1 |
|- ( <. a , t >. = <. <. x , y >. , z >. -> a = <. x , y >. ) |
10 |
6 9
|
syl6bi |
|- ( w = <. a , t >. -> ( w = <. <. x , y >. , z >. -> a = <. x , y >. ) ) |
11 |
|
vex |
|- x e. _V |
12 |
|
vex |
|- y e. _V |
13 |
11 12
|
eqvinop |
|- ( a = <. x , y >. <-> E. r E. s ( a = <. r , s >. /\ <. r , s >. = <. x , y >. ) ) |
14 |
|
opeq1 |
|- ( a = <. r , s >. -> <. a , t >. = <. <. r , s >. , t >. ) |
15 |
14
|
eqeq2d |
|- ( a = <. r , s >. -> ( w = <. a , t >. <-> w = <. <. r , s >. , t >. ) ) |
16 |
11 12 3
|
otth2 |
|- ( <. <. x , y >. , z >. = <. <. r , s >. , t >. <-> ( x = r /\ y = s /\ z = t ) ) |
17 |
|
euequ |
|- E! x x = r |
18 |
|
eupick |
|- ( ( E! x x = r /\ E. x ( x = r /\ E. y ( y = s /\ E. z ( z = t /\ ph ) ) ) ) -> ( x = r -> E. y ( y = s /\ E. z ( z = t /\ ph ) ) ) ) |
19 |
17 18
|
mpan |
|- ( E. x ( x = r /\ E. y ( y = s /\ E. z ( z = t /\ ph ) ) ) -> ( x = r -> E. y ( y = s /\ E. z ( z = t /\ ph ) ) ) ) |
20 |
|
euequ |
|- E! y y = s |
21 |
|
eupick |
|- ( ( E! y y = s /\ E. y ( y = s /\ E. z ( z = t /\ ph ) ) ) -> ( y = s -> E. z ( z = t /\ ph ) ) ) |
22 |
20 21
|
mpan |
|- ( E. y ( y = s /\ E. z ( z = t /\ ph ) ) -> ( y = s -> E. z ( z = t /\ ph ) ) ) |
23 |
|
euequ |
|- E! z z = t |
24 |
|
eupick |
|- ( ( E! z z = t /\ E. z ( z = t /\ ph ) ) -> ( z = t -> ph ) ) |
25 |
23 24
|
mpan |
|- ( E. z ( z = t /\ ph ) -> ( z = t -> ph ) ) |
26 |
22 25
|
syl6 |
|- ( E. y ( y = s /\ E. z ( z = t /\ ph ) ) -> ( y = s -> ( z = t -> ph ) ) ) |
27 |
19 26
|
syl6 |
|- ( E. x ( x = r /\ E. y ( y = s /\ E. z ( z = t /\ ph ) ) ) -> ( x = r -> ( y = s -> ( z = t -> ph ) ) ) ) |
28 |
27
|
3impd |
|- ( E. x ( x = r /\ E. y ( y = s /\ E. z ( z = t /\ ph ) ) ) -> ( ( x = r /\ y = s /\ z = t ) -> ph ) ) |
29 |
16 28
|
syl5bi |
|- ( E. x ( x = r /\ E. y ( y = s /\ E. z ( z = t /\ ph ) ) ) -> ( <. <. x , y >. , z >. = <. <. r , s >. , t >. -> ph ) ) |
30 |
|
df-3an |
|- ( ( x = r /\ y = s /\ z = t ) <-> ( ( x = r /\ y = s ) /\ z = t ) ) |
31 |
16 30
|
bitri |
|- ( <. <. x , y >. , z >. = <. <. r , s >. , t >. <-> ( ( x = r /\ y = s ) /\ z = t ) ) |
32 |
31
|
anbi1i |
|- ( ( <. <. x , y >. , z >. = <. <. r , s >. , t >. /\ ph ) <-> ( ( ( x = r /\ y = s ) /\ z = t ) /\ ph ) ) |
33 |
|
anass |
|- ( ( ( ( x = r /\ y = s ) /\ z = t ) /\ ph ) <-> ( ( x = r /\ y = s ) /\ ( z = t /\ ph ) ) ) |
34 |
|
anass |
|- ( ( ( x = r /\ y = s ) /\ ( z = t /\ ph ) ) <-> ( x = r /\ ( y = s /\ ( z = t /\ ph ) ) ) ) |
35 |
32 33 34
|
3bitri |
|- ( ( <. <. x , y >. , z >. = <. <. r , s >. , t >. /\ ph ) <-> ( x = r /\ ( y = s /\ ( z = t /\ ph ) ) ) ) |
36 |
35
|
3exbii |
|- ( E. x E. y E. z ( <. <. x , y >. , z >. = <. <. r , s >. , t >. /\ ph ) <-> E. x E. y E. z ( x = r /\ ( y = s /\ ( z = t /\ ph ) ) ) ) |
37 |
|
nfe1 |
|- F/ x E. x ( x = r /\ E. z ( y = s /\ ( z = t /\ ph ) ) ) |
38 |
|
19.8a |
|- ( ( y = s /\ ( z = t /\ ph ) ) -> E. z ( y = s /\ ( z = t /\ ph ) ) ) |
39 |
38
|
anim2i |
|- ( ( x = r /\ ( y = s /\ ( z = t /\ ph ) ) ) -> ( x = r /\ E. z ( y = s /\ ( z = t /\ ph ) ) ) ) |
40 |
39
|
eximi |
|- ( E. z ( x = r /\ ( y = s /\ ( z = t /\ ph ) ) ) -> E. z ( x = r /\ E. z ( y = s /\ ( z = t /\ ph ) ) ) ) |
41 |
|
biidd |
|- ( A. x x = z -> ( ( x = r /\ E. z ( y = s /\ ( z = t /\ ph ) ) ) <-> ( x = r /\ E. z ( y = s /\ ( z = t /\ ph ) ) ) ) ) |
42 |
41
|
drex1v |
|- ( A. x x = z -> ( E. x ( x = r /\ E. z ( y = s /\ ( z = t /\ ph ) ) ) <-> E. z ( x = r /\ E. z ( y = s /\ ( z = t /\ ph ) ) ) ) ) |
43 |
40 42
|
syl5ibr |
|- ( A. x x = z -> ( E. z ( x = r /\ ( y = s /\ ( z = t /\ ph ) ) ) -> E. x ( x = r /\ E. z ( y = s /\ ( z = t /\ ph ) ) ) ) ) |
44 |
|
19.40 |
|- ( E. z ( x = r /\ ( y = s /\ ( z = t /\ ph ) ) ) -> ( E. z x = r /\ E. z ( y = s /\ ( z = t /\ ph ) ) ) ) |
45 |
|
nfvd |
|- ( -. A. x x = z -> F/ z x = r ) |
46 |
45
|
19.9d |
|- ( -. A. x x = z -> ( E. z x = r -> x = r ) ) |
47 |
46
|
anim1d |
|- ( -. A. x x = z -> ( ( E. z x = r /\ E. z ( y = s /\ ( z = t /\ ph ) ) ) -> ( x = r /\ E. z ( y = s /\ ( z = t /\ ph ) ) ) ) ) |
48 |
|
19.8a |
|- ( ( x = r /\ E. z ( y = s /\ ( z = t /\ ph ) ) ) -> E. x ( x = r /\ E. z ( y = s /\ ( z = t /\ ph ) ) ) ) |
49 |
44 47 48
|
syl56 |
|- ( -. A. x x = z -> ( E. z ( x = r /\ ( y = s /\ ( z = t /\ ph ) ) ) -> E. x ( x = r /\ E. z ( y = s /\ ( z = t /\ ph ) ) ) ) ) |
50 |
43 49
|
pm2.61i |
|- ( E. z ( x = r /\ ( y = s /\ ( z = t /\ ph ) ) ) -> E. x ( x = r /\ E. z ( y = s /\ ( z = t /\ ph ) ) ) ) |
51 |
37 50
|
exlimi |
|- ( E. x E. z ( x = r /\ ( y = s /\ ( z = t /\ ph ) ) ) -> E. x ( x = r /\ E. z ( y = s /\ ( z = t /\ ph ) ) ) ) |
52 |
51
|
eximi |
|- ( E. y E. x E. z ( x = r /\ ( y = s /\ ( z = t /\ ph ) ) ) -> E. y E. x ( x = r /\ E. z ( y = s /\ ( z = t /\ ph ) ) ) ) |
53 |
|
excom |
|- ( E. x E. y E. z ( x = r /\ ( y = s /\ ( z = t /\ ph ) ) ) <-> E. y E. x E. z ( x = r /\ ( y = s /\ ( z = t /\ ph ) ) ) ) |
54 |
|
excom |
|- ( E. x E. y ( x = r /\ E. z ( y = s /\ ( z = t /\ ph ) ) ) <-> E. y E. x ( x = r /\ E. z ( y = s /\ ( z = t /\ ph ) ) ) ) |
55 |
52 53 54
|
3imtr4i |
|- ( E. x E. y E. z ( x = r /\ ( y = s /\ ( z = t /\ ph ) ) ) -> E. x E. y ( x = r /\ E. z ( y = s /\ ( z = t /\ ph ) ) ) ) |
56 |
|
nfe1 |
|- F/ x E. x ( x = r /\ E. y E. z ( y = s /\ ( z = t /\ ph ) ) ) |
57 |
|
19.8a |
|- ( E. z ( y = s /\ ( z = t /\ ph ) ) -> E. y E. z ( y = s /\ ( z = t /\ ph ) ) ) |
58 |
57
|
anim2i |
|- ( ( x = r /\ E. z ( y = s /\ ( z = t /\ ph ) ) ) -> ( x = r /\ E. y E. z ( y = s /\ ( z = t /\ ph ) ) ) ) |
59 |
58
|
eximi |
|- ( E. y ( x = r /\ E. z ( y = s /\ ( z = t /\ ph ) ) ) -> E. y ( x = r /\ E. y E. z ( y = s /\ ( z = t /\ ph ) ) ) ) |
60 |
|
biidd |
|- ( A. x x = y -> ( ( x = r /\ E. y E. z ( y = s /\ ( z = t /\ ph ) ) ) <-> ( x = r /\ E. y E. z ( y = s /\ ( z = t /\ ph ) ) ) ) ) |
61 |
60
|
drex1v |
|- ( A. x x = y -> ( E. x ( x = r /\ E. y E. z ( y = s /\ ( z = t /\ ph ) ) ) <-> E. y ( x = r /\ E. y E. z ( y = s /\ ( z = t /\ ph ) ) ) ) ) |
62 |
59 61
|
syl5ibr |
|- ( A. x x = y -> ( E. y ( x = r /\ E. z ( y = s /\ ( z = t /\ ph ) ) ) -> E. x ( x = r /\ E. y E. z ( y = s /\ ( z = t /\ ph ) ) ) ) ) |
63 |
|
19.40 |
|- ( E. y ( x = r /\ E. z ( y = s /\ ( z = t /\ ph ) ) ) -> ( E. y x = r /\ E. y E. z ( y = s /\ ( z = t /\ ph ) ) ) ) |
64 |
|
nfvd |
|- ( -. A. x x = y -> F/ y x = r ) |
65 |
64
|
19.9d |
|- ( -. A. x x = y -> ( E. y x = r -> x = r ) ) |
66 |
65
|
anim1d |
|- ( -. A. x x = y -> ( ( E. y x = r /\ E. y E. z ( y = s /\ ( z = t /\ ph ) ) ) -> ( x = r /\ E. y E. z ( y = s /\ ( z = t /\ ph ) ) ) ) ) |
67 |
|
19.8a |
|- ( ( x = r /\ E. y E. z ( y = s /\ ( z = t /\ ph ) ) ) -> E. x ( x = r /\ E. y E. z ( y = s /\ ( z = t /\ ph ) ) ) ) |
68 |
63 66 67
|
syl56 |
|- ( -. A. x x = y -> ( E. y ( x = r /\ E. z ( y = s /\ ( z = t /\ ph ) ) ) -> E. x ( x = r /\ E. y E. z ( y = s /\ ( z = t /\ ph ) ) ) ) ) |
69 |
62 68
|
pm2.61i |
|- ( E. y ( x = r /\ E. z ( y = s /\ ( z = t /\ ph ) ) ) -> E. x ( x = r /\ E. y E. z ( y = s /\ ( z = t /\ ph ) ) ) ) |
70 |
56 69
|
exlimi |
|- ( E. x E. y ( x = r /\ E. z ( y = s /\ ( z = t /\ ph ) ) ) -> E. x ( x = r /\ E. y E. z ( y = s /\ ( z = t /\ ph ) ) ) ) |
71 |
|
nfe1 |
|- F/ y E. y ( y = s /\ E. z ( z = t /\ ph ) ) |
72 |
|
19.8a |
|- ( ( z = t /\ ph ) -> E. z ( z = t /\ ph ) ) |
73 |
72
|
anim2i |
|- ( ( y = s /\ ( z = t /\ ph ) ) -> ( y = s /\ E. z ( z = t /\ ph ) ) ) |
74 |
73
|
eximi |
|- ( E. z ( y = s /\ ( z = t /\ ph ) ) -> E. z ( y = s /\ E. z ( z = t /\ ph ) ) ) |
75 |
|
biidd |
|- ( A. y y = z -> ( ( y = s /\ E. z ( z = t /\ ph ) ) <-> ( y = s /\ E. z ( z = t /\ ph ) ) ) ) |
76 |
75
|
drex1v |
|- ( A. y y = z -> ( E. y ( y = s /\ E. z ( z = t /\ ph ) ) <-> E. z ( y = s /\ E. z ( z = t /\ ph ) ) ) ) |
77 |
74 76
|
syl5ibr |
|- ( A. y y = z -> ( E. z ( y = s /\ ( z = t /\ ph ) ) -> E. y ( y = s /\ E. z ( z = t /\ ph ) ) ) ) |
78 |
|
19.40 |
|- ( E. z ( y = s /\ ( z = t /\ ph ) ) -> ( E. z y = s /\ E. z ( z = t /\ ph ) ) ) |
79 |
|
nfvd |
|- ( -. A. y y = z -> F/ z y = s ) |
80 |
79
|
19.9d |
|- ( -. A. y y = z -> ( E. z y = s -> y = s ) ) |
81 |
80
|
anim1d |
|- ( -. A. y y = z -> ( ( E. z y = s /\ E. z ( z = t /\ ph ) ) -> ( y = s /\ E. z ( z = t /\ ph ) ) ) ) |
82 |
|
19.8a |
|- ( ( y = s /\ E. z ( z = t /\ ph ) ) -> E. y ( y = s /\ E. z ( z = t /\ ph ) ) ) |
83 |
78 81 82
|
syl56 |
|- ( -. A. y y = z -> ( E. z ( y = s /\ ( z = t /\ ph ) ) -> E. y ( y = s /\ E. z ( z = t /\ ph ) ) ) ) |
84 |
77 83
|
pm2.61i |
|- ( E. z ( y = s /\ ( z = t /\ ph ) ) -> E. y ( y = s /\ E. z ( z = t /\ ph ) ) ) |
85 |
71 84
|
exlimi |
|- ( E. y E. z ( y = s /\ ( z = t /\ ph ) ) -> E. y ( y = s /\ E. z ( z = t /\ ph ) ) ) |
86 |
85
|
anim2i |
|- ( ( x = r /\ E. y E. z ( y = s /\ ( z = t /\ ph ) ) ) -> ( x = r /\ E. y ( y = s /\ E. z ( z = t /\ ph ) ) ) ) |
87 |
86
|
eximi |
|- ( E. x ( x = r /\ E. y E. z ( y = s /\ ( z = t /\ ph ) ) ) -> E. x ( x = r /\ E. y ( y = s /\ E. z ( z = t /\ ph ) ) ) ) |
88 |
55 70 87
|
3syl |
|- ( E. x E. y E. z ( x = r /\ ( y = s /\ ( z = t /\ ph ) ) ) -> E. x ( x = r /\ E. y ( y = s /\ E. z ( z = t /\ ph ) ) ) ) |
89 |
36 88
|
sylbi |
|- ( E. x E. y E. z ( <. <. x , y >. , z >. = <. <. r , s >. , t >. /\ ph ) -> E. x ( x = r /\ E. y ( y = s /\ E. z ( z = t /\ ph ) ) ) ) |
90 |
29 89
|
syl11 |
|- ( <. <. x , y >. , z >. = <. <. r , s >. , t >. -> ( E. x E. y E. z ( <. <. x , y >. , z >. = <. <. r , s >. , t >. /\ ph ) -> ph ) ) |
91 |
|
eqeq1 |
|- ( w = <. <. r , s >. , t >. -> ( w = <. <. x , y >. , z >. <-> <. <. r , s >. , t >. = <. <. x , y >. , z >. ) ) |
92 |
|
eqcom |
|- ( <. <. r , s >. , t >. = <. <. x , y >. , z >. <-> <. <. x , y >. , z >. = <. <. r , s >. , t >. ) |
93 |
91 92
|
bitrdi |
|- ( w = <. <. r , s >. , t >. -> ( w = <. <. x , y >. , z >. <-> <. <. x , y >. , z >. = <. <. r , s >. , t >. ) ) |
94 |
93
|
anbi1d |
|- ( w = <. <. r , s >. , t >. -> ( ( w = <. <. x , y >. , z >. /\ ph ) <-> ( <. <. x , y >. , z >. = <. <. r , s >. , t >. /\ ph ) ) ) |
95 |
94
|
3exbidv |
|- ( w = <. <. r , s >. , t >. -> ( E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) <-> E. x E. y E. z ( <. <. x , y >. , z >. = <. <. r , s >. , t >. /\ ph ) ) ) |
96 |
95
|
imbi1d |
|- ( w = <. <. r , s >. , t >. -> ( ( E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) -> ph ) <-> ( E. x E. y E. z ( <. <. x , y >. , z >. = <. <. r , s >. , t >. /\ ph ) -> ph ) ) ) |
97 |
93 96
|
imbi12d |
|- ( w = <. <. r , s >. , t >. -> ( ( w = <. <. x , y >. , z >. -> ( E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) -> ph ) ) <-> ( <. <. x , y >. , z >. = <. <. r , s >. , t >. -> ( E. x E. y E. z ( <. <. x , y >. , z >. = <. <. r , s >. , t >. /\ ph ) -> ph ) ) ) ) |
98 |
90 97
|
mpbiri |
|- ( w = <. <. r , s >. , t >. -> ( w = <. <. x , y >. , z >. -> ( E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) -> ph ) ) ) |
99 |
15 98
|
syl6bi |
|- ( a = <. r , s >. -> ( w = <. a , t >. -> ( w = <. <. x , y >. , z >. -> ( E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) -> ph ) ) ) ) |
100 |
99
|
adantr |
|- ( ( a = <. r , s >. /\ <. r , s >. = <. x , y >. ) -> ( w = <. a , t >. -> ( w = <. <. x , y >. , z >. -> ( E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) -> ph ) ) ) ) |
101 |
100
|
exlimivv |
|- ( E. r E. s ( a = <. r , s >. /\ <. r , s >. = <. x , y >. ) -> ( w = <. a , t >. -> ( w = <. <. x , y >. , z >. -> ( E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) -> ph ) ) ) ) |
102 |
13 101
|
sylbi |
|- ( a = <. x , y >. -> ( w = <. a , t >. -> ( w = <. <. x , y >. , z >. -> ( E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) -> ph ) ) ) ) |
103 |
102
|
com3l |
|- ( w = <. a , t >. -> ( w = <. <. x , y >. , z >. -> ( a = <. x , y >. -> ( E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) -> ph ) ) ) ) |
104 |
10 103
|
mpdd |
|- ( w = <. a , t >. -> ( w = <. <. x , y >. , z >. -> ( E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) -> ph ) ) ) |
105 |
104
|
adantr |
|- ( ( w = <. a , t >. /\ <. a , t >. = <. <. x , y >. , z >. ) -> ( w = <. <. x , y >. , z >. -> ( E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) -> ph ) ) ) |
106 |
105
|
exlimivv |
|- ( E. a E. t ( w = <. a , t >. /\ <. a , t >. = <. <. x , y >. , z >. ) -> ( w = <. <. x , y >. , z >. -> ( E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) -> ph ) ) ) |
107 |
5 106
|
mpcom |
|- ( w = <. <. x , y >. , z >. -> ( E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) -> ph ) ) |
108 |
|
19.8a |
|- ( ( w = <. <. x , y >. , z >. /\ ph ) -> E. z ( w = <. <. x , y >. , z >. /\ ph ) ) |
109 |
|
19.8a |
|- ( E. z ( w = <. <. x , y >. , z >. /\ ph ) -> E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) ) |
110 |
|
19.8a |
|- ( E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) -> E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) ) |
111 |
108 109 110
|
3syl |
|- ( ( w = <. <. x , y >. , z >. /\ ph ) -> E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) ) |
112 |
111
|
ex |
|- ( w = <. <. x , y >. , z >. -> ( ph -> E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) ) ) |
113 |
107 112
|
impbid |
|- ( w = <. <. x , y >. , z >. -> ( E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) <-> ph ) ) |
114 |
|
df-oprab |
|- { <. <. x , y >. , z >. | ph } = { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) } |
115 |
1 113 114
|
elab2 |
|- ( <. <. x , y >. , z >. e. { <. <. x , y >. , z >. | ph } <-> ph ) |