Step |
Hyp |
Ref |
Expression |
1 |
|
df-9 |
|- 9 = ( 8 + 1 ) |
2 |
|
8even |
|- 8 e. Even |
3 |
|
evenp1odd |
|- ( 8 e. Even -> ( 8 + 1 ) e. Odd ) |
4 |
2 3
|
ax-mp |
|- ( 8 + 1 ) e. Odd |
5 |
1 4
|
eqeltri |
|- 9 e. Odd |
6 |
|
3prm |
|- 3 e. Prime |
7 |
|
3odd |
|- 3 e. Odd |
8 |
7 7 7
|
3pm3.2i |
|- ( 3 e. Odd /\ 3 e. Odd /\ 3 e. Odd ) |
9 |
|
gbpart9 |
|- 9 = ( ( 3 + 3 ) + 3 ) |
10 |
8 9
|
pm3.2i |
|- ( ( 3 e. Odd /\ 3 e. Odd /\ 3 e. Odd ) /\ 9 = ( ( 3 + 3 ) + 3 ) ) |
11 |
|
eleq1 |
|- ( r = 3 -> ( r e. Odd <-> 3 e. Odd ) ) |
12 |
11
|
3anbi3d |
|- ( r = 3 -> ( ( 3 e. Odd /\ 3 e. Odd /\ r e. Odd ) <-> ( 3 e. Odd /\ 3 e. Odd /\ 3 e. Odd ) ) ) |
13 |
|
oveq2 |
|- ( r = 3 -> ( ( 3 + 3 ) + r ) = ( ( 3 + 3 ) + 3 ) ) |
14 |
13
|
eqeq2d |
|- ( r = 3 -> ( 9 = ( ( 3 + 3 ) + r ) <-> 9 = ( ( 3 + 3 ) + 3 ) ) ) |
15 |
12 14
|
anbi12d |
|- ( r = 3 -> ( ( ( 3 e. Odd /\ 3 e. Odd /\ r e. Odd ) /\ 9 = ( ( 3 + 3 ) + r ) ) <-> ( ( 3 e. Odd /\ 3 e. Odd /\ 3 e. Odd ) /\ 9 = ( ( 3 + 3 ) + 3 ) ) ) ) |
16 |
15
|
rspcev |
|- ( ( 3 e. Prime /\ ( ( 3 e. Odd /\ 3 e. Odd /\ 3 e. Odd ) /\ 9 = ( ( 3 + 3 ) + 3 ) ) ) -> E. r e. Prime ( ( 3 e. Odd /\ 3 e. Odd /\ r e. Odd ) /\ 9 = ( ( 3 + 3 ) + r ) ) ) |
17 |
6 10 16
|
mp2an |
|- E. r e. Prime ( ( 3 e. Odd /\ 3 e. Odd /\ r e. Odd ) /\ 9 = ( ( 3 + 3 ) + r ) ) |
18 |
|
eleq1 |
|- ( p = 3 -> ( p e. Odd <-> 3 e. Odd ) ) |
19 |
18
|
3anbi1d |
|- ( p = 3 -> ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) <-> ( 3 e. Odd /\ q e. Odd /\ r e. Odd ) ) ) |
20 |
|
oveq1 |
|- ( p = 3 -> ( p + q ) = ( 3 + q ) ) |
21 |
20
|
oveq1d |
|- ( p = 3 -> ( ( p + q ) + r ) = ( ( 3 + q ) + r ) ) |
22 |
21
|
eqeq2d |
|- ( p = 3 -> ( 9 = ( ( p + q ) + r ) <-> 9 = ( ( 3 + q ) + r ) ) ) |
23 |
19 22
|
anbi12d |
|- ( p = 3 -> ( ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) /\ 9 = ( ( p + q ) + r ) ) <-> ( ( 3 e. Odd /\ q e. Odd /\ r e. Odd ) /\ 9 = ( ( 3 + q ) + r ) ) ) ) |
24 |
23
|
rexbidv |
|- ( p = 3 -> ( E. r e. Prime ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) /\ 9 = ( ( p + q ) + r ) ) <-> E. r e. Prime ( ( 3 e. Odd /\ q e. Odd /\ r e. Odd ) /\ 9 = ( ( 3 + q ) + r ) ) ) ) |
25 |
|
eleq1 |
|- ( q = 3 -> ( q e. Odd <-> 3 e. Odd ) ) |
26 |
25
|
3anbi2d |
|- ( q = 3 -> ( ( 3 e. Odd /\ q e. Odd /\ r e. Odd ) <-> ( 3 e. Odd /\ 3 e. Odd /\ r e. Odd ) ) ) |
27 |
|
oveq2 |
|- ( q = 3 -> ( 3 + q ) = ( 3 + 3 ) ) |
28 |
27
|
oveq1d |
|- ( q = 3 -> ( ( 3 + q ) + r ) = ( ( 3 + 3 ) + r ) ) |
29 |
28
|
eqeq2d |
|- ( q = 3 -> ( 9 = ( ( 3 + q ) + r ) <-> 9 = ( ( 3 + 3 ) + r ) ) ) |
30 |
26 29
|
anbi12d |
|- ( q = 3 -> ( ( ( 3 e. Odd /\ q e. Odd /\ r e. Odd ) /\ 9 = ( ( 3 + q ) + r ) ) <-> ( ( 3 e. Odd /\ 3 e. Odd /\ r e. Odd ) /\ 9 = ( ( 3 + 3 ) + r ) ) ) ) |
31 |
30
|
rexbidv |
|- ( q = 3 -> ( E. r e. Prime ( ( 3 e. Odd /\ q e. Odd /\ r e. Odd ) /\ 9 = ( ( 3 + q ) + r ) ) <-> E. r e. Prime ( ( 3 e. Odd /\ 3 e. Odd /\ r e. Odd ) /\ 9 = ( ( 3 + 3 ) + r ) ) ) ) |
32 |
24 31
|
rspc2ev |
|- ( ( 3 e. Prime /\ 3 e. Prime /\ E. r e. Prime ( ( 3 e. Odd /\ 3 e. Odd /\ r e. Odd ) /\ 9 = ( ( 3 + 3 ) + r ) ) ) -> E. p e. Prime E. q e. Prime E. r e. Prime ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) /\ 9 = ( ( p + q ) + r ) ) ) |
33 |
6 6 17 32
|
mp3an |
|- E. p e. Prime E. q e. Prime E. r e. Prime ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) /\ 9 = ( ( p + q ) + r ) ) |
34 |
|
isgbo |
|- ( 9 e. GoldbachOdd <-> ( 9 e. Odd /\ E. p e. Prime E. q e. Prime E. r e. Prime ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) /\ 9 = ( ( p + q ) + r ) ) ) ) |
35 |
5 33 34
|
mpbir2an |
|- 9 e. GoldbachOdd |