Step |
Hyp |
Ref |
Expression |
1 |
|
disjxpin.1 |
|- ( x = ( 1st ` p ) -> C = E ) |
2 |
|
disjxpin.2 |
|- ( y = ( 2nd ` p ) -> D = F ) |
3 |
|
disjxpin.3 |
|- ( ph -> Disj_ x e. A C ) |
4 |
|
disjxpin.4 |
|- ( ph -> Disj_ y e. B D ) |
5 |
|
xp1st |
|- ( q e. ( A X. B ) -> ( 1st ` q ) e. A ) |
6 |
5
|
ad2antrl |
|- ( ( ph /\ ( q e. ( A X. B ) /\ r e. ( A X. B ) ) ) -> ( 1st ` q ) e. A ) |
7 |
|
xp1st |
|- ( r e. ( A X. B ) -> ( 1st ` r ) e. A ) |
8 |
7
|
ad2antll |
|- ( ( ph /\ ( q e. ( A X. B ) /\ r e. ( A X. B ) ) ) -> ( 1st ` r ) e. A ) |
9 |
|
simpl |
|- ( ( ph /\ ( q e. ( A X. B ) /\ r e. ( A X. B ) ) ) -> ph ) |
10 |
|
disjors |
|- ( Disj_ x e. A C <-> A. a e. A A. c e. A ( a = c \/ ( [_ a / x ]_ C i^i [_ c / x ]_ C ) = (/) ) ) |
11 |
3 10
|
sylib |
|- ( ph -> A. a e. A A. c e. A ( a = c \/ ( [_ a / x ]_ C i^i [_ c / x ]_ C ) = (/) ) ) |
12 |
|
eqeq1 |
|- ( a = ( 1st ` q ) -> ( a = c <-> ( 1st ` q ) = c ) ) |
13 |
|
csbeq1 |
|- ( a = ( 1st ` q ) -> [_ a / x ]_ C = [_ ( 1st ` q ) / x ]_ C ) |
14 |
13
|
ineq1d |
|- ( a = ( 1st ` q ) -> ( [_ a / x ]_ C i^i [_ c / x ]_ C ) = ( [_ ( 1st ` q ) / x ]_ C i^i [_ c / x ]_ C ) ) |
15 |
14
|
eqeq1d |
|- ( a = ( 1st ` q ) -> ( ( [_ a / x ]_ C i^i [_ c / x ]_ C ) = (/) <-> ( [_ ( 1st ` q ) / x ]_ C i^i [_ c / x ]_ C ) = (/) ) ) |
16 |
12 15
|
orbi12d |
|- ( a = ( 1st ` q ) -> ( ( a = c \/ ( [_ a / x ]_ C i^i [_ c / x ]_ C ) = (/) ) <-> ( ( 1st ` q ) = c \/ ( [_ ( 1st ` q ) / x ]_ C i^i [_ c / x ]_ C ) = (/) ) ) ) |
17 |
|
eqeq2 |
|- ( c = ( 1st ` r ) -> ( ( 1st ` q ) = c <-> ( 1st ` q ) = ( 1st ` r ) ) ) |
18 |
|
csbeq1 |
|- ( c = ( 1st ` r ) -> [_ c / x ]_ C = [_ ( 1st ` r ) / x ]_ C ) |
19 |
18
|
ineq2d |
|- ( c = ( 1st ` r ) -> ( [_ ( 1st ` q ) / x ]_ C i^i [_ c / x ]_ C ) = ( [_ ( 1st ` q ) / x ]_ C i^i [_ ( 1st ` r ) / x ]_ C ) ) |
20 |
19
|
eqeq1d |
|- ( c = ( 1st ` r ) -> ( ( [_ ( 1st ` q ) / x ]_ C i^i [_ c / x ]_ C ) = (/) <-> ( [_ ( 1st ` q ) / x ]_ C i^i [_ ( 1st ` r ) / x ]_ C ) = (/) ) ) |
21 |
17 20
|
orbi12d |
|- ( c = ( 1st ` r ) -> ( ( ( 1st ` q ) = c \/ ( [_ ( 1st ` q ) / x ]_ C i^i [_ c / x ]_ C ) = (/) ) <-> ( ( 1st ` q ) = ( 1st ` r ) \/ ( [_ ( 1st ` q ) / x ]_ C i^i [_ ( 1st ` r ) / x ]_ C ) = (/) ) ) ) |
22 |
16 21
|
rspc2v |
|- ( ( ( 1st ` q ) e. A /\ ( 1st ` r ) e. A ) -> ( A. a e. A A. c e. A ( a = c \/ ( [_ a / x ]_ C i^i [_ c / x ]_ C ) = (/) ) -> ( ( 1st ` q ) = ( 1st ` r ) \/ ( [_ ( 1st ` q ) / x ]_ C i^i [_ ( 1st ` r ) / x ]_ C ) = (/) ) ) ) |
23 |
11 22
|
syl5 |
|- ( ( ( 1st ` q ) e. A /\ ( 1st ` r ) e. A ) -> ( ph -> ( ( 1st ` q ) = ( 1st ` r ) \/ ( [_ ( 1st ` q ) / x ]_ C i^i [_ ( 1st ` r ) / x ]_ C ) = (/) ) ) ) |
24 |
23
|
imp |
|- ( ( ( ( 1st ` q ) e. A /\ ( 1st ` r ) e. A ) /\ ph ) -> ( ( 1st ` q ) = ( 1st ` r ) \/ ( [_ ( 1st ` q ) / x ]_ C i^i [_ ( 1st ` r ) / x ]_ C ) = (/) ) ) |
25 |
6 8 9 24
|
syl21anc |
|- ( ( ph /\ ( q e. ( A X. B ) /\ r e. ( A X. B ) ) ) -> ( ( 1st ` q ) = ( 1st ` r ) \/ ( [_ ( 1st ` q ) / x ]_ C i^i [_ ( 1st ` r ) / x ]_ C ) = (/) ) ) |
26 |
|
xp2nd |
|- ( q e. ( A X. B ) -> ( 2nd ` q ) e. B ) |
27 |
26
|
ad2antrl |
|- ( ( ph /\ ( q e. ( A X. B ) /\ r e. ( A X. B ) ) ) -> ( 2nd ` q ) e. B ) |
28 |
|
xp2nd |
|- ( r e. ( A X. B ) -> ( 2nd ` r ) e. B ) |
29 |
28
|
ad2antll |
|- ( ( ph /\ ( q e. ( A X. B ) /\ r e. ( A X. B ) ) ) -> ( 2nd ` r ) e. B ) |
30 |
|
disjors |
|- ( Disj_ y e. B D <-> A. b e. B A. d e. B ( b = d \/ ( [_ b / y ]_ D i^i [_ d / y ]_ D ) = (/) ) ) |
31 |
4 30
|
sylib |
|- ( ph -> A. b e. B A. d e. B ( b = d \/ ( [_ b / y ]_ D i^i [_ d / y ]_ D ) = (/) ) ) |
32 |
|
eqeq1 |
|- ( b = ( 2nd ` q ) -> ( b = d <-> ( 2nd ` q ) = d ) ) |
33 |
|
csbeq1 |
|- ( b = ( 2nd ` q ) -> [_ b / y ]_ D = [_ ( 2nd ` q ) / y ]_ D ) |
34 |
33
|
ineq1d |
|- ( b = ( 2nd ` q ) -> ( [_ b / y ]_ D i^i [_ d / y ]_ D ) = ( [_ ( 2nd ` q ) / y ]_ D i^i [_ d / y ]_ D ) ) |
35 |
34
|
eqeq1d |
|- ( b = ( 2nd ` q ) -> ( ( [_ b / y ]_ D i^i [_ d / y ]_ D ) = (/) <-> ( [_ ( 2nd ` q ) / y ]_ D i^i [_ d / y ]_ D ) = (/) ) ) |
36 |
32 35
|
orbi12d |
|- ( b = ( 2nd ` q ) -> ( ( b = d \/ ( [_ b / y ]_ D i^i [_ d / y ]_ D ) = (/) ) <-> ( ( 2nd ` q ) = d \/ ( [_ ( 2nd ` q ) / y ]_ D i^i [_ d / y ]_ D ) = (/) ) ) ) |
37 |
|
eqeq2 |
|- ( d = ( 2nd ` r ) -> ( ( 2nd ` q ) = d <-> ( 2nd ` q ) = ( 2nd ` r ) ) ) |
38 |
|
csbeq1 |
|- ( d = ( 2nd ` r ) -> [_ d / y ]_ D = [_ ( 2nd ` r ) / y ]_ D ) |
39 |
38
|
ineq2d |
|- ( d = ( 2nd ` r ) -> ( [_ ( 2nd ` q ) / y ]_ D i^i [_ d / y ]_ D ) = ( [_ ( 2nd ` q ) / y ]_ D i^i [_ ( 2nd ` r ) / y ]_ D ) ) |
40 |
39
|
eqeq1d |
|- ( d = ( 2nd ` r ) -> ( ( [_ ( 2nd ` q ) / y ]_ D i^i [_ d / y ]_ D ) = (/) <-> ( [_ ( 2nd ` q ) / y ]_ D i^i [_ ( 2nd ` r ) / y ]_ D ) = (/) ) ) |
41 |
37 40
|
orbi12d |
|- ( d = ( 2nd ` r ) -> ( ( ( 2nd ` q ) = d \/ ( [_ ( 2nd ` q ) / y ]_ D i^i [_ d / y ]_ D ) = (/) ) <-> ( ( 2nd ` q ) = ( 2nd ` r ) \/ ( [_ ( 2nd ` q ) / y ]_ D i^i [_ ( 2nd ` r ) / y ]_ D ) = (/) ) ) ) |
42 |
36 41
|
rspc2v |
|- ( ( ( 2nd ` q ) e. B /\ ( 2nd ` r ) e. B ) -> ( A. b e. B A. d e. B ( b = d \/ ( [_ b / y ]_ D i^i [_ d / y ]_ D ) = (/) ) -> ( ( 2nd ` q ) = ( 2nd ` r ) \/ ( [_ ( 2nd ` q ) / y ]_ D i^i [_ ( 2nd ` r ) / y ]_ D ) = (/) ) ) ) |
43 |
31 42
|
syl5 |
|- ( ( ( 2nd ` q ) e. B /\ ( 2nd ` r ) e. B ) -> ( ph -> ( ( 2nd ` q ) = ( 2nd ` r ) \/ ( [_ ( 2nd ` q ) / y ]_ D i^i [_ ( 2nd ` r ) / y ]_ D ) = (/) ) ) ) |
44 |
43
|
imp |
|- ( ( ( ( 2nd ` q ) e. B /\ ( 2nd ` r ) e. B ) /\ ph ) -> ( ( 2nd ` q ) = ( 2nd ` r ) \/ ( [_ ( 2nd ` q ) / y ]_ D i^i [_ ( 2nd ` r ) / y ]_ D ) = (/) ) ) |
45 |
27 29 9 44
|
syl21anc |
|- ( ( ph /\ ( q e. ( A X. B ) /\ r e. ( A X. B ) ) ) -> ( ( 2nd ` q ) = ( 2nd ` r ) \/ ( [_ ( 2nd ` q ) / y ]_ D i^i [_ ( 2nd ` r ) / y ]_ D ) = (/) ) ) |
46 |
25 45
|
jca |
|- ( ( ph /\ ( q e. ( A X. B ) /\ r e. ( A X. B ) ) ) -> ( ( ( 1st ` q ) = ( 1st ` r ) \/ ( [_ ( 1st ` q ) / x ]_ C i^i [_ ( 1st ` r ) / x ]_ C ) = (/) ) /\ ( ( 2nd ` q ) = ( 2nd ` r ) \/ ( [_ ( 2nd ` q ) / y ]_ D i^i [_ ( 2nd ` r ) / y ]_ D ) = (/) ) ) ) |
47 |
|
anddi |
|- ( ( ( ( 1st ` q ) = ( 1st ` r ) \/ ( [_ ( 1st ` q ) / x ]_ C i^i [_ ( 1st ` r ) / x ]_ C ) = (/) ) /\ ( ( 2nd ` q ) = ( 2nd ` r ) \/ ( [_ ( 2nd ` q ) / y ]_ D i^i [_ ( 2nd ` r ) / y ]_ D ) = (/) ) ) <-> ( ( ( ( 1st ` q ) = ( 1st ` r ) /\ ( 2nd ` q ) = ( 2nd ` r ) ) \/ ( ( 1st ` q ) = ( 1st ` r ) /\ ( [_ ( 2nd ` q ) / y ]_ D i^i [_ ( 2nd ` r ) / y ]_ D ) = (/) ) ) \/ ( ( ( [_ ( 1st ` q ) / x ]_ C i^i [_ ( 1st ` r ) / x ]_ C ) = (/) /\ ( 2nd ` q ) = ( 2nd ` r ) ) \/ ( ( [_ ( 1st ` q ) / x ]_ C i^i [_ ( 1st ` r ) / x ]_ C ) = (/) /\ ( [_ ( 2nd ` q ) / y ]_ D i^i [_ ( 2nd ` r ) / y ]_ D ) = (/) ) ) ) ) |
48 |
46 47
|
sylib |
|- ( ( ph /\ ( q e. ( A X. B ) /\ r e. ( A X. B ) ) ) -> ( ( ( ( 1st ` q ) = ( 1st ` r ) /\ ( 2nd ` q ) = ( 2nd ` r ) ) \/ ( ( 1st ` q ) = ( 1st ` r ) /\ ( [_ ( 2nd ` q ) / y ]_ D i^i [_ ( 2nd ` r ) / y ]_ D ) = (/) ) ) \/ ( ( ( [_ ( 1st ` q ) / x ]_ C i^i [_ ( 1st ` r ) / x ]_ C ) = (/) /\ ( 2nd ` q ) = ( 2nd ` r ) ) \/ ( ( [_ ( 1st ` q ) / x ]_ C i^i [_ ( 1st ` r ) / x ]_ C ) = (/) /\ ( [_ ( 2nd ` q ) / y ]_ D i^i [_ ( 2nd ` r ) / y ]_ D ) = (/) ) ) ) ) |
49 |
|
orass |
|- ( ( ( ( ( 1st ` q ) = ( 1st ` r ) /\ ( 2nd ` q ) = ( 2nd ` r ) ) \/ ( ( 1st ` q ) = ( 1st ` r ) /\ ( [_ ( 2nd ` q ) / y ]_ D i^i [_ ( 2nd ` r ) / y ]_ D ) = (/) ) ) \/ ( ( ( [_ ( 1st ` q ) / x ]_ C i^i [_ ( 1st ` r ) / x ]_ C ) = (/) /\ ( 2nd ` q ) = ( 2nd ` r ) ) \/ ( ( [_ ( 1st ` q ) / x ]_ C i^i [_ ( 1st ` r ) / x ]_ C ) = (/) /\ ( [_ ( 2nd ` q ) / y ]_ D i^i [_ ( 2nd ` r ) / y ]_ D ) = (/) ) ) ) <-> ( ( ( 1st ` q ) = ( 1st ` r ) /\ ( 2nd ` q ) = ( 2nd ` r ) ) \/ ( ( ( 1st ` q ) = ( 1st ` r ) /\ ( [_ ( 2nd ` q ) / y ]_ D i^i [_ ( 2nd ` r ) / y ]_ D ) = (/) ) \/ ( ( ( [_ ( 1st ` q ) / x ]_ C i^i [_ ( 1st ` r ) / x ]_ C ) = (/) /\ ( 2nd ` q ) = ( 2nd ` r ) ) \/ ( ( [_ ( 1st ` q ) / x ]_ C i^i [_ ( 1st ` r ) / x ]_ C ) = (/) /\ ( [_ ( 2nd ` q ) / y ]_ D i^i [_ ( 2nd ` r ) / y ]_ D ) = (/) ) ) ) ) ) |
50 |
48 49
|
sylib |
|- ( ( ph /\ ( q e. ( A X. B ) /\ r e. ( A X. B ) ) ) -> ( ( ( 1st ` q ) = ( 1st ` r ) /\ ( 2nd ` q ) = ( 2nd ` r ) ) \/ ( ( ( 1st ` q ) = ( 1st ` r ) /\ ( [_ ( 2nd ` q ) / y ]_ D i^i [_ ( 2nd ` r ) / y ]_ D ) = (/) ) \/ ( ( ( [_ ( 1st ` q ) / x ]_ C i^i [_ ( 1st ` r ) / x ]_ C ) = (/) /\ ( 2nd ` q ) = ( 2nd ` r ) ) \/ ( ( [_ ( 1st ` q ) / x ]_ C i^i [_ ( 1st ` r ) / x ]_ C ) = (/) /\ ( [_ ( 2nd ` q ) / y ]_ D i^i [_ ( 2nd ` r ) / y ]_ D ) = (/) ) ) ) ) ) |
51 |
|
xpopth |
|- ( ( q e. ( A X. B ) /\ r e. ( A X. B ) ) -> ( ( ( 1st ` q ) = ( 1st ` r ) /\ ( 2nd ` q ) = ( 2nd ` r ) ) <-> q = r ) ) |
52 |
51
|
adantl |
|- ( ( ph /\ ( q e. ( A X. B ) /\ r e. ( A X. B ) ) ) -> ( ( ( 1st ` q ) = ( 1st ` r ) /\ ( 2nd ` q ) = ( 2nd ` r ) ) <-> q = r ) ) |
53 |
52
|
biimpd |
|- ( ( ph /\ ( q e. ( A X. B ) /\ r e. ( A X. B ) ) ) -> ( ( ( 1st ` q ) = ( 1st ` r ) /\ ( 2nd ` q ) = ( 2nd ` r ) ) -> q = r ) ) |
54 |
|
inss2 |
|- ( ( [_ q / p ]_ E i^i [_ r / p ]_ E ) i^i ( [_ q / p ]_ F i^i [_ r / p ]_ F ) ) C_ ( [_ q / p ]_ F i^i [_ r / p ]_ F ) |
55 |
|
csbin |
|- [_ q / p ]_ ( E i^i F ) = ( [_ q / p ]_ E i^i [_ q / p ]_ F ) |
56 |
|
csbin |
|- [_ r / p ]_ ( E i^i F ) = ( [_ r / p ]_ E i^i [_ r / p ]_ F ) |
57 |
55 56
|
ineq12i |
|- ( [_ q / p ]_ ( E i^i F ) i^i [_ r / p ]_ ( E i^i F ) ) = ( ( [_ q / p ]_ E i^i [_ q / p ]_ F ) i^i ( [_ r / p ]_ E i^i [_ r / p ]_ F ) ) |
58 |
|
in4 |
|- ( ( [_ q / p ]_ E i^i [_ q / p ]_ F ) i^i ( [_ r / p ]_ E i^i [_ r / p ]_ F ) ) = ( ( [_ q / p ]_ E i^i [_ r / p ]_ E ) i^i ( [_ q / p ]_ F i^i [_ r / p ]_ F ) ) |
59 |
57 58
|
eqtri |
|- ( [_ q / p ]_ ( E i^i F ) i^i [_ r / p ]_ ( E i^i F ) ) = ( ( [_ q / p ]_ E i^i [_ r / p ]_ E ) i^i ( [_ q / p ]_ F i^i [_ r / p ]_ F ) ) |
60 |
|
vex |
|- q e. _V |
61 |
|
csbnestgw |
|- ( q e. _V -> [_ q / p ]_ [_ ( 2nd ` p ) / y ]_ D = [_ [_ q / p ]_ ( 2nd ` p ) / y ]_ D ) |
62 |
60 61
|
ax-mp |
|- [_ q / p ]_ [_ ( 2nd ` p ) / y ]_ D = [_ [_ q / p ]_ ( 2nd ` p ) / y ]_ D |
63 |
|
fvex |
|- ( 2nd ` p ) e. _V |
64 |
63 2
|
csbie |
|- [_ ( 2nd ` p ) / y ]_ D = F |
65 |
64
|
csbeq2i |
|- [_ q / p ]_ [_ ( 2nd ` p ) / y ]_ D = [_ q / p ]_ F |
66 |
|
csbfv |
|- [_ q / p ]_ ( 2nd ` p ) = ( 2nd ` q ) |
67 |
|
csbeq1 |
|- ( [_ q / p ]_ ( 2nd ` p ) = ( 2nd ` q ) -> [_ [_ q / p ]_ ( 2nd ` p ) / y ]_ D = [_ ( 2nd ` q ) / y ]_ D ) |
68 |
66 67
|
ax-mp |
|- [_ [_ q / p ]_ ( 2nd ` p ) / y ]_ D = [_ ( 2nd ` q ) / y ]_ D |
69 |
62 65 68
|
3eqtr3ri |
|- [_ ( 2nd ` q ) / y ]_ D = [_ q / p ]_ F |
70 |
|
vex |
|- r e. _V |
71 |
|
csbnestgw |
|- ( r e. _V -> [_ r / p ]_ [_ ( 2nd ` p ) / y ]_ D = [_ [_ r / p ]_ ( 2nd ` p ) / y ]_ D ) |
72 |
70 71
|
ax-mp |
|- [_ r / p ]_ [_ ( 2nd ` p ) / y ]_ D = [_ [_ r / p ]_ ( 2nd ` p ) / y ]_ D |
73 |
64
|
csbeq2i |
|- [_ r / p ]_ [_ ( 2nd ` p ) / y ]_ D = [_ r / p ]_ F |
74 |
|
csbfv |
|- [_ r / p ]_ ( 2nd ` p ) = ( 2nd ` r ) |
75 |
|
csbeq1 |
|- ( [_ r / p ]_ ( 2nd ` p ) = ( 2nd ` r ) -> [_ [_ r / p ]_ ( 2nd ` p ) / y ]_ D = [_ ( 2nd ` r ) / y ]_ D ) |
76 |
74 75
|
ax-mp |
|- [_ [_ r / p ]_ ( 2nd ` p ) / y ]_ D = [_ ( 2nd ` r ) / y ]_ D |
77 |
72 73 76
|
3eqtr3ri |
|- [_ ( 2nd ` r ) / y ]_ D = [_ r / p ]_ F |
78 |
69 77
|
ineq12i |
|- ( [_ ( 2nd ` q ) / y ]_ D i^i [_ ( 2nd ` r ) / y ]_ D ) = ( [_ q / p ]_ F i^i [_ r / p ]_ F ) |
79 |
54 59 78
|
3sstr4i |
|- ( [_ q / p ]_ ( E i^i F ) i^i [_ r / p ]_ ( E i^i F ) ) C_ ( [_ ( 2nd ` q ) / y ]_ D i^i [_ ( 2nd ` r ) / y ]_ D ) |
80 |
|
sseq0 |
|- ( ( ( [_ q / p ]_ ( E i^i F ) i^i [_ r / p ]_ ( E i^i F ) ) C_ ( [_ ( 2nd ` q ) / y ]_ D i^i [_ ( 2nd ` r ) / y ]_ D ) /\ ( [_ ( 2nd ` q ) / y ]_ D i^i [_ ( 2nd ` r ) / y ]_ D ) = (/) ) -> ( [_ q / p ]_ ( E i^i F ) i^i [_ r / p ]_ ( E i^i F ) ) = (/) ) |
81 |
79 80
|
mpan |
|- ( ( [_ ( 2nd ` q ) / y ]_ D i^i [_ ( 2nd ` r ) / y ]_ D ) = (/) -> ( [_ q / p ]_ ( E i^i F ) i^i [_ r / p ]_ ( E i^i F ) ) = (/) ) |
82 |
81
|
a1i |
|- ( ( ph /\ ( q e. ( A X. B ) /\ r e. ( A X. B ) ) ) -> ( ( [_ ( 2nd ` q ) / y ]_ D i^i [_ ( 2nd ` r ) / y ]_ D ) = (/) -> ( [_ q / p ]_ ( E i^i F ) i^i [_ r / p ]_ ( E i^i F ) ) = (/) ) ) |
83 |
82
|
adantld |
|- ( ( ph /\ ( q e. ( A X. B ) /\ r e. ( A X. B ) ) ) -> ( ( ( 1st ` q ) = ( 1st ` r ) /\ ( [_ ( 2nd ` q ) / y ]_ D i^i [_ ( 2nd ` r ) / y ]_ D ) = (/) ) -> ( [_ q / p ]_ ( E i^i F ) i^i [_ r / p ]_ ( E i^i F ) ) = (/) ) ) |
84 |
|
inss1 |
|- ( ( [_ q / p ]_ E i^i [_ r / p ]_ E ) i^i ( [_ q / p ]_ F i^i [_ r / p ]_ F ) ) C_ ( [_ q / p ]_ E i^i [_ r / p ]_ E ) |
85 |
|
csbnestgw |
|- ( q e. _V -> [_ q / p ]_ [_ ( 1st ` p ) / x ]_ C = [_ [_ q / p ]_ ( 1st ` p ) / x ]_ C ) |
86 |
60 85
|
ax-mp |
|- [_ q / p ]_ [_ ( 1st ` p ) / x ]_ C = [_ [_ q / p ]_ ( 1st ` p ) / x ]_ C |
87 |
|
fvex |
|- ( 1st ` p ) e. _V |
88 |
87 1
|
csbie |
|- [_ ( 1st ` p ) / x ]_ C = E |
89 |
88
|
csbeq2i |
|- [_ q / p ]_ [_ ( 1st ` p ) / x ]_ C = [_ q / p ]_ E |
90 |
|
csbfv |
|- [_ q / p ]_ ( 1st ` p ) = ( 1st ` q ) |
91 |
|
csbeq1 |
|- ( [_ q / p ]_ ( 1st ` p ) = ( 1st ` q ) -> [_ [_ q / p ]_ ( 1st ` p ) / x ]_ C = [_ ( 1st ` q ) / x ]_ C ) |
92 |
90 91
|
ax-mp |
|- [_ [_ q / p ]_ ( 1st ` p ) / x ]_ C = [_ ( 1st ` q ) / x ]_ C |
93 |
86 89 92
|
3eqtr3ri |
|- [_ ( 1st ` q ) / x ]_ C = [_ q / p ]_ E |
94 |
|
csbnestgw |
|- ( r e. _V -> [_ r / p ]_ [_ ( 1st ` p ) / x ]_ C = [_ [_ r / p ]_ ( 1st ` p ) / x ]_ C ) |
95 |
70 94
|
ax-mp |
|- [_ r / p ]_ [_ ( 1st ` p ) / x ]_ C = [_ [_ r / p ]_ ( 1st ` p ) / x ]_ C |
96 |
88
|
csbeq2i |
|- [_ r / p ]_ [_ ( 1st ` p ) / x ]_ C = [_ r / p ]_ E |
97 |
|
csbfv |
|- [_ r / p ]_ ( 1st ` p ) = ( 1st ` r ) |
98 |
|
csbeq1 |
|- ( [_ r / p ]_ ( 1st ` p ) = ( 1st ` r ) -> [_ [_ r / p ]_ ( 1st ` p ) / x ]_ C = [_ ( 1st ` r ) / x ]_ C ) |
99 |
97 98
|
ax-mp |
|- [_ [_ r / p ]_ ( 1st ` p ) / x ]_ C = [_ ( 1st ` r ) / x ]_ C |
100 |
95 96 99
|
3eqtr3ri |
|- [_ ( 1st ` r ) / x ]_ C = [_ r / p ]_ E |
101 |
93 100
|
ineq12i |
|- ( [_ ( 1st ` q ) / x ]_ C i^i [_ ( 1st ` r ) / x ]_ C ) = ( [_ q / p ]_ E i^i [_ r / p ]_ E ) |
102 |
84 59 101
|
3sstr4i |
|- ( [_ q / p ]_ ( E i^i F ) i^i [_ r / p ]_ ( E i^i F ) ) C_ ( [_ ( 1st ` q ) / x ]_ C i^i [_ ( 1st ` r ) / x ]_ C ) |
103 |
|
sseq0 |
|- ( ( ( [_ q / p ]_ ( E i^i F ) i^i [_ r / p ]_ ( E i^i F ) ) C_ ( [_ ( 1st ` q ) / x ]_ C i^i [_ ( 1st ` r ) / x ]_ C ) /\ ( [_ ( 1st ` q ) / x ]_ C i^i [_ ( 1st ` r ) / x ]_ C ) = (/) ) -> ( [_ q / p ]_ ( E i^i F ) i^i [_ r / p ]_ ( E i^i F ) ) = (/) ) |
104 |
102 103
|
mpan |
|- ( ( [_ ( 1st ` q ) / x ]_ C i^i [_ ( 1st ` r ) / x ]_ C ) = (/) -> ( [_ q / p ]_ ( E i^i F ) i^i [_ r / p ]_ ( E i^i F ) ) = (/) ) |
105 |
104
|
a1i |
|- ( ( ph /\ ( q e. ( A X. B ) /\ r e. ( A X. B ) ) ) -> ( ( [_ ( 1st ` q ) / x ]_ C i^i [_ ( 1st ` r ) / x ]_ C ) = (/) -> ( [_ q / p ]_ ( E i^i F ) i^i [_ r / p ]_ ( E i^i F ) ) = (/) ) ) |
106 |
105
|
adantrd |
|- ( ( ph /\ ( q e. ( A X. B ) /\ r e. ( A X. B ) ) ) -> ( ( ( [_ ( 1st ` q ) / x ]_ C i^i [_ ( 1st ` r ) / x ]_ C ) = (/) /\ ( 2nd ` q ) = ( 2nd ` r ) ) -> ( [_ q / p ]_ ( E i^i F ) i^i [_ r / p ]_ ( E i^i F ) ) = (/) ) ) |
107 |
82
|
adantld |
|- ( ( ph /\ ( q e. ( A X. B ) /\ r e. ( A X. B ) ) ) -> ( ( ( [_ ( 1st ` q ) / x ]_ C i^i [_ ( 1st ` r ) / x ]_ C ) = (/) /\ ( [_ ( 2nd ` q ) / y ]_ D i^i [_ ( 2nd ` r ) / y ]_ D ) = (/) ) -> ( [_ q / p ]_ ( E i^i F ) i^i [_ r / p ]_ ( E i^i F ) ) = (/) ) ) |
108 |
106 107
|
jaod |
|- ( ( ph /\ ( q e. ( A X. B ) /\ r e. ( A X. B ) ) ) -> ( ( ( ( [_ ( 1st ` q ) / x ]_ C i^i [_ ( 1st ` r ) / x ]_ C ) = (/) /\ ( 2nd ` q ) = ( 2nd ` r ) ) \/ ( ( [_ ( 1st ` q ) / x ]_ C i^i [_ ( 1st ` r ) / x ]_ C ) = (/) /\ ( [_ ( 2nd ` q ) / y ]_ D i^i [_ ( 2nd ` r ) / y ]_ D ) = (/) ) ) -> ( [_ q / p ]_ ( E i^i F ) i^i [_ r / p ]_ ( E i^i F ) ) = (/) ) ) |
109 |
83 108
|
jaod |
|- ( ( ph /\ ( q e. ( A X. B ) /\ r e. ( A X. B ) ) ) -> ( ( ( ( 1st ` q ) = ( 1st ` r ) /\ ( [_ ( 2nd ` q ) / y ]_ D i^i [_ ( 2nd ` r ) / y ]_ D ) = (/) ) \/ ( ( ( [_ ( 1st ` q ) / x ]_ C i^i [_ ( 1st ` r ) / x ]_ C ) = (/) /\ ( 2nd ` q ) = ( 2nd ` r ) ) \/ ( ( [_ ( 1st ` q ) / x ]_ C i^i [_ ( 1st ` r ) / x ]_ C ) = (/) /\ ( [_ ( 2nd ` q ) / y ]_ D i^i [_ ( 2nd ` r ) / y ]_ D ) = (/) ) ) ) -> ( [_ q / p ]_ ( E i^i F ) i^i [_ r / p ]_ ( E i^i F ) ) = (/) ) ) |
110 |
53 109
|
orim12d |
|- ( ( ph /\ ( q e. ( A X. B ) /\ r e. ( A X. B ) ) ) -> ( ( ( ( 1st ` q ) = ( 1st ` r ) /\ ( 2nd ` q ) = ( 2nd ` r ) ) \/ ( ( ( 1st ` q ) = ( 1st ` r ) /\ ( [_ ( 2nd ` q ) / y ]_ D i^i [_ ( 2nd ` r ) / y ]_ D ) = (/) ) \/ ( ( ( [_ ( 1st ` q ) / x ]_ C i^i [_ ( 1st ` r ) / x ]_ C ) = (/) /\ ( 2nd ` q ) = ( 2nd ` r ) ) \/ ( ( [_ ( 1st ` q ) / x ]_ C i^i [_ ( 1st ` r ) / x ]_ C ) = (/) /\ ( [_ ( 2nd ` q ) / y ]_ D i^i [_ ( 2nd ` r ) / y ]_ D ) = (/) ) ) ) ) -> ( q = r \/ ( [_ q / p ]_ ( E i^i F ) i^i [_ r / p ]_ ( E i^i F ) ) = (/) ) ) ) |
111 |
50 110
|
mpd |
|- ( ( ph /\ ( q e. ( A X. B ) /\ r e. ( A X. B ) ) ) -> ( q = r \/ ( [_ q / p ]_ ( E i^i F ) i^i [_ r / p ]_ ( E i^i F ) ) = (/) ) ) |
112 |
111
|
ralrimivva |
|- ( ph -> A. q e. ( A X. B ) A. r e. ( A X. B ) ( q = r \/ ( [_ q / p ]_ ( E i^i F ) i^i [_ r / p ]_ ( E i^i F ) ) = (/) ) ) |
113 |
|
disjors |
|- ( Disj_ p e. ( A X. B ) ( E i^i F ) <-> A. q e. ( A X. B ) A. r e. ( A X. B ) ( q = r \/ ( [_ q / p ]_ ( E i^i F ) i^i [_ r / p ]_ ( E i^i F ) ) = (/) ) ) |
114 |
112 113
|
sylibr |
|- ( ph -> Disj_ p e. ( A X. B ) ( E i^i F ) ) |