Step |
Hyp |
Ref |
Expression |
1 |
|
6p5e11 |
|- ( 6 + 5 ) = ; 1 1 |
2 |
|
6even |
|- 6 e. Even |
3 |
|
5odd |
|- 5 e. Odd |
4 |
|
epoo |
|- ( ( 6 e. Even /\ 5 e. Odd ) -> ( 6 + 5 ) e. Odd ) |
5 |
2 3 4
|
mp2an |
|- ( 6 + 5 ) e. Odd |
6 |
1 5
|
eqeltrri |
|- ; 1 1 e. Odd |
7 |
|
3prm |
|- 3 e. Prime |
8 |
|
5prm |
|- 5 e. Prime |
9 |
|
3odd |
|- 3 e. Odd |
10 |
9 9 3
|
3pm3.2i |
|- ( 3 e. Odd /\ 3 e. Odd /\ 5 e. Odd ) |
11 |
|
gbpart11 |
|- ; 1 1 = ( ( 3 + 3 ) + 5 ) |
12 |
10 11
|
pm3.2i |
|- ( ( 3 e. Odd /\ 3 e. Odd /\ 5 e. Odd ) /\ ; 1 1 = ( ( 3 + 3 ) + 5 ) ) |
13 |
|
eleq1 |
|- ( r = 5 -> ( r e. Odd <-> 5 e. Odd ) ) |
14 |
13
|
3anbi3d |
|- ( r = 5 -> ( ( 3 e. Odd /\ 3 e. Odd /\ r e. Odd ) <-> ( 3 e. Odd /\ 3 e. Odd /\ 5 e. Odd ) ) ) |
15 |
|
oveq2 |
|- ( r = 5 -> ( ( 3 + 3 ) + r ) = ( ( 3 + 3 ) + 5 ) ) |
16 |
15
|
eqeq2d |
|- ( r = 5 -> ( ; 1 1 = ( ( 3 + 3 ) + r ) <-> ; 1 1 = ( ( 3 + 3 ) + 5 ) ) ) |
17 |
14 16
|
anbi12d |
|- ( r = 5 -> ( ( ( 3 e. Odd /\ 3 e. Odd /\ r e. Odd ) /\ ; 1 1 = ( ( 3 + 3 ) + r ) ) <-> ( ( 3 e. Odd /\ 3 e. Odd /\ 5 e. Odd ) /\ ; 1 1 = ( ( 3 + 3 ) + 5 ) ) ) ) |
18 |
17
|
rspcev |
|- ( ( 5 e. Prime /\ ( ( 3 e. Odd /\ 3 e. Odd /\ 5 e. Odd ) /\ ; 1 1 = ( ( 3 + 3 ) + 5 ) ) ) -> E. r e. Prime ( ( 3 e. Odd /\ 3 e. Odd /\ r e. Odd ) /\ ; 1 1 = ( ( 3 + 3 ) + r ) ) ) |
19 |
8 12 18
|
mp2an |
|- E. r e. Prime ( ( 3 e. Odd /\ 3 e. Odd /\ r e. Odd ) /\ ; 1 1 = ( ( 3 + 3 ) + r ) ) |
20 |
|
eleq1 |
|- ( p = 3 -> ( p e. Odd <-> 3 e. Odd ) ) |
21 |
20
|
3anbi1d |
|- ( p = 3 -> ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) <-> ( 3 e. Odd /\ q e. Odd /\ r e. Odd ) ) ) |
22 |
|
oveq1 |
|- ( p = 3 -> ( p + q ) = ( 3 + q ) ) |
23 |
22
|
oveq1d |
|- ( p = 3 -> ( ( p + q ) + r ) = ( ( 3 + q ) + r ) ) |
24 |
23
|
eqeq2d |
|- ( p = 3 -> ( ; 1 1 = ( ( p + q ) + r ) <-> ; 1 1 = ( ( 3 + q ) + r ) ) ) |
25 |
21 24
|
anbi12d |
|- ( p = 3 -> ( ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) /\ ; 1 1 = ( ( p + q ) + r ) ) <-> ( ( 3 e. Odd /\ q e. Odd /\ r e. Odd ) /\ ; 1 1 = ( ( 3 + q ) + r ) ) ) ) |
26 |
25
|
rexbidv |
|- ( p = 3 -> ( E. r e. Prime ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) /\ ; 1 1 = ( ( p + q ) + r ) ) <-> E. r e. Prime ( ( 3 e. Odd /\ q e. Odd /\ r e. Odd ) /\ ; 1 1 = ( ( 3 + q ) + r ) ) ) ) |
27 |
|
eleq1 |
|- ( q = 3 -> ( q e. Odd <-> 3 e. Odd ) ) |
28 |
27
|
3anbi2d |
|- ( q = 3 -> ( ( 3 e. Odd /\ q e. Odd /\ r e. Odd ) <-> ( 3 e. Odd /\ 3 e. Odd /\ r e. Odd ) ) ) |
29 |
|
oveq2 |
|- ( q = 3 -> ( 3 + q ) = ( 3 + 3 ) ) |
30 |
29
|
oveq1d |
|- ( q = 3 -> ( ( 3 + q ) + r ) = ( ( 3 + 3 ) + r ) ) |
31 |
30
|
eqeq2d |
|- ( q = 3 -> ( ; 1 1 = ( ( 3 + q ) + r ) <-> ; 1 1 = ( ( 3 + 3 ) + r ) ) ) |
32 |
28 31
|
anbi12d |
|- ( q = 3 -> ( ( ( 3 e. Odd /\ q e. Odd /\ r e. Odd ) /\ ; 1 1 = ( ( 3 + q ) + r ) ) <-> ( ( 3 e. Odd /\ 3 e. Odd /\ r e. Odd ) /\ ; 1 1 = ( ( 3 + 3 ) + r ) ) ) ) |
33 |
32
|
rexbidv |
|- ( q = 3 -> ( E. r e. Prime ( ( 3 e. Odd /\ q e. Odd /\ r e. Odd ) /\ ; 1 1 = ( ( 3 + q ) + r ) ) <-> E. r e. Prime ( ( 3 e. Odd /\ 3 e. Odd /\ r e. Odd ) /\ ; 1 1 = ( ( 3 + 3 ) + r ) ) ) ) |
34 |
26 33
|
rspc2ev |
|- ( ( 3 e. Prime /\ 3 e. Prime /\ E. r e. Prime ( ( 3 e. Odd /\ 3 e. Odd /\ r e. Odd ) /\ ; 1 1 = ( ( 3 + 3 ) + r ) ) ) -> E. p e. Prime E. q e. Prime E. r e. Prime ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) /\ ; 1 1 = ( ( p + q ) + r ) ) ) |
35 |
7 7 19 34
|
mp3an |
|- E. p e. Prime E. q e. Prime E. r e. Prime ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) /\ ; 1 1 = ( ( p + q ) + r ) ) |
36 |
|
isgbo |
|- ( ; 1 1 e. GoldbachOdd <-> ( ; 1 1 e. Odd /\ E. p e. Prime E. q e. Prime E. r e. Prime ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) /\ ; 1 1 = ( ( p + q ) + r ) ) ) ) |
37 |
6 35 36
|
mpbir2an |
|- ; 1 1 e. GoldbachOdd |