Step |
Hyp |
Ref |
Expression |
1 |
|
df-9 |
⊢ 9 = ( 8 + 1 ) |
2 |
|
8even |
⊢ 8 ∈ Even |
3 |
|
evenp1odd |
⊢ ( 8 ∈ Even → ( 8 + 1 ) ∈ Odd ) |
4 |
2 3
|
ax-mp |
⊢ ( 8 + 1 ) ∈ Odd |
5 |
1 4
|
eqeltri |
⊢ 9 ∈ Odd |
6 |
|
3prm |
⊢ 3 ∈ ℙ |
7 |
|
3odd |
⊢ 3 ∈ Odd |
8 |
7 7 7
|
3pm3.2i |
⊢ ( 3 ∈ Odd ∧ 3 ∈ Odd ∧ 3 ∈ Odd ) |
9 |
|
gbpart9 |
⊢ 9 = ( ( 3 + 3 ) + 3 ) |
10 |
8 9
|
pm3.2i |
⊢ ( ( 3 ∈ Odd ∧ 3 ∈ Odd ∧ 3 ∈ Odd ) ∧ 9 = ( ( 3 + 3 ) + 3 ) ) |
11 |
|
eleq1 |
⊢ ( 𝑟 = 3 → ( 𝑟 ∈ Odd ↔ 3 ∈ Odd ) ) |
12 |
11
|
3anbi3d |
⊢ ( 𝑟 = 3 → ( ( 3 ∈ Odd ∧ 3 ∈ Odd ∧ 𝑟 ∈ Odd ) ↔ ( 3 ∈ Odd ∧ 3 ∈ Odd ∧ 3 ∈ 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 ∈ Odd ∧ 3 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 9 = ( ( 3 + 3 ) + 𝑟 ) ) ↔ ( ( 3 ∈ Odd ∧ 3 ∈ Odd ∧ 3 ∈ Odd ) ∧ 9 = ( ( 3 + 3 ) + 3 ) ) ) ) |
16 |
15
|
rspcev |
⊢ ( ( 3 ∈ ℙ ∧ ( ( 3 ∈ Odd ∧ 3 ∈ Odd ∧ 3 ∈ Odd ) ∧ 9 = ( ( 3 + 3 ) + 3 ) ) ) → ∃ 𝑟 ∈ ℙ ( ( 3 ∈ Odd ∧ 3 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 9 = ( ( 3 + 3 ) + 𝑟 ) ) ) |
17 |
6 10 16
|
mp2an |
⊢ ∃ 𝑟 ∈ ℙ ( ( 3 ∈ Odd ∧ 3 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 9 = ( ( 3 + 3 ) + 𝑟 ) ) |
18 |
|
eleq1 |
⊢ ( 𝑝 = 3 → ( 𝑝 ∈ Odd ↔ 3 ∈ Odd ) ) |
19 |
18
|
3anbi1d |
⊢ ( 𝑝 = 3 → ( ( 𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ↔ ( 3 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ 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 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 9 = ( ( 𝑝 + 𝑞 ) + 𝑟 ) ) ↔ ( ( 3 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 9 = ( ( 3 + 𝑞 ) + 𝑟 ) ) ) ) |
24 |
23
|
rexbidv |
⊢ ( 𝑝 = 3 → ( ∃ 𝑟 ∈ ℙ ( ( 𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 9 = ( ( 𝑝 + 𝑞 ) + 𝑟 ) ) ↔ ∃ 𝑟 ∈ ℙ ( ( 3 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 9 = ( ( 3 + 𝑞 ) + 𝑟 ) ) ) ) |
25 |
|
eleq1 |
⊢ ( 𝑞 = 3 → ( 𝑞 ∈ Odd ↔ 3 ∈ Odd ) ) |
26 |
25
|
3anbi2d |
⊢ ( 𝑞 = 3 → ( ( 3 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ↔ ( 3 ∈ Odd ∧ 3 ∈ Odd ∧ 𝑟 ∈ 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 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 9 = ( ( 3 + 𝑞 ) + 𝑟 ) ) ↔ ( ( 3 ∈ Odd ∧ 3 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 9 = ( ( 3 + 3 ) + 𝑟 ) ) ) ) |
31 |
30
|
rexbidv |
⊢ ( 𝑞 = 3 → ( ∃ 𝑟 ∈ ℙ ( ( 3 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 9 = ( ( 3 + 𝑞 ) + 𝑟 ) ) ↔ ∃ 𝑟 ∈ ℙ ( ( 3 ∈ Odd ∧ 3 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 9 = ( ( 3 + 3 ) + 𝑟 ) ) ) ) |
32 |
24 31
|
rspc2ev |
⊢ ( ( 3 ∈ ℙ ∧ 3 ∈ ℙ ∧ ∃ 𝑟 ∈ ℙ ( ( 3 ∈ Odd ∧ 3 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 9 = ( ( 3 + 3 ) + 𝑟 ) ) ) → ∃ 𝑝 ∈ ℙ ∃ 𝑞 ∈ ℙ ∃ 𝑟 ∈ ℙ ( ( 𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 9 = ( ( 𝑝 + 𝑞 ) + 𝑟 ) ) ) |
33 |
6 6 17 32
|
mp3an |
⊢ ∃ 𝑝 ∈ ℙ ∃ 𝑞 ∈ ℙ ∃ 𝑟 ∈ ℙ ( ( 𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 9 = ( ( 𝑝 + 𝑞 ) + 𝑟 ) ) |
34 |
|
isgbo |
⊢ ( 9 ∈ GoldbachOdd ↔ ( 9 ∈ Odd ∧ ∃ 𝑝 ∈ ℙ ∃ 𝑞 ∈ ℙ ∃ 𝑟 ∈ ℙ ( ( 𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 9 = ( ( 𝑝 + 𝑞 ) + 𝑟 ) ) ) ) |
35 |
5 33 34
|
mpbir2an |
⊢ 9 ∈ GoldbachOdd |