Step |
Hyp |
Ref |
Expression |
1 |
|
8even |
|- 8 e. Even |
2 |
|
5prm |
|- 5 e. Prime |
3 |
|
3prm |
|- 3 e. Prime |
4 |
|
5odd |
|- 5 e. Odd |
5 |
|
3odd |
|- 3 e. Odd |
6 |
|
5p3e8 |
|- ( 5 + 3 ) = 8 |
7 |
6
|
eqcomi |
|- 8 = ( 5 + 3 ) |
8 |
4 5 7
|
3pm3.2i |
|- ( 5 e. Odd /\ 3 e. Odd /\ 8 = ( 5 + 3 ) ) |
9 |
|
eleq1 |
|- ( p = 5 -> ( p e. Odd <-> 5 e. Odd ) ) |
10 |
|
biidd |
|- ( p = 5 -> ( q e. Odd <-> q e. Odd ) ) |
11 |
|
oveq1 |
|- ( p = 5 -> ( p + q ) = ( 5 + q ) ) |
12 |
11
|
eqeq2d |
|- ( p = 5 -> ( 8 = ( p + q ) <-> 8 = ( 5 + q ) ) ) |
13 |
9 10 12
|
3anbi123d |
|- ( p = 5 -> ( ( p e. Odd /\ q e. Odd /\ 8 = ( p + q ) ) <-> ( 5 e. Odd /\ q e. Odd /\ 8 = ( 5 + q ) ) ) ) |
14 |
|
biidd |
|- ( q = 3 -> ( 5 e. Odd <-> 5 e. Odd ) ) |
15 |
|
eleq1 |
|- ( q = 3 -> ( q e. Odd <-> 3 e. Odd ) ) |
16 |
|
oveq2 |
|- ( q = 3 -> ( 5 + q ) = ( 5 + 3 ) ) |
17 |
16
|
eqeq2d |
|- ( q = 3 -> ( 8 = ( 5 + q ) <-> 8 = ( 5 + 3 ) ) ) |
18 |
14 15 17
|
3anbi123d |
|- ( q = 3 -> ( ( 5 e. Odd /\ q e. Odd /\ 8 = ( 5 + q ) ) <-> ( 5 e. Odd /\ 3 e. Odd /\ 8 = ( 5 + 3 ) ) ) ) |
19 |
13 18
|
rspc2ev |
|- ( ( 5 e. Prime /\ 3 e. Prime /\ ( 5 e. Odd /\ 3 e. Odd /\ 8 = ( 5 + 3 ) ) ) -> E. p e. Prime E. q e. Prime ( p e. Odd /\ q e. Odd /\ 8 = ( p + q ) ) ) |
20 |
2 3 8 19
|
mp3an |
|- E. p e. Prime E. q e. Prime ( p e. Odd /\ q e. Odd /\ 8 = ( p + q ) ) |
21 |
|
isgbe |
|- ( 8 e. GoldbachEven <-> ( 8 e. Even /\ E. p e. Prime E. q e. Prime ( p e. Odd /\ q e. Odd /\ 8 = ( p + q ) ) ) ) |
22 |
1 20 21
|
mpbir2an |
|- 8 e. GoldbachEven |