Step |
Hyp |
Ref |
Expression |
1 |
|
7odd |
⊢ 7 ∈ Odd |
2 |
|
2prm |
⊢ 2 ∈ ℙ |
3 |
|
3prm |
⊢ 3 ∈ ℙ |
4 |
|
gbpart7 |
⊢ 7 = ( ( 2 + 2 ) + 3 ) |
5 |
|
oveq2 |
⊢ ( 𝑟 = 3 → ( ( 2 + 2 ) + 𝑟 ) = ( ( 2 + 2 ) + 3 ) ) |
6 |
5
|
rspceeqv |
⊢ ( ( 3 ∈ ℙ ∧ 7 = ( ( 2 + 2 ) + 3 ) ) → ∃ 𝑟 ∈ ℙ 7 = ( ( 2 + 2 ) + 𝑟 ) ) |
7 |
3 4 6
|
mp2an |
⊢ ∃ 𝑟 ∈ ℙ 7 = ( ( 2 + 2 ) + 𝑟 ) |
8 |
|
oveq1 |
⊢ ( 𝑝 = 2 → ( 𝑝 + 𝑞 ) = ( 2 + 𝑞 ) ) |
9 |
8
|
oveq1d |
⊢ ( 𝑝 = 2 → ( ( 𝑝 + 𝑞 ) + 𝑟 ) = ( ( 2 + 𝑞 ) + 𝑟 ) ) |
10 |
9
|
eqeq2d |
⊢ ( 𝑝 = 2 → ( 7 = ( ( 𝑝 + 𝑞 ) + 𝑟 ) ↔ 7 = ( ( 2 + 𝑞 ) + 𝑟 ) ) ) |
11 |
10
|
rexbidv |
⊢ ( 𝑝 = 2 → ( ∃ 𝑟 ∈ ℙ 7 = ( ( 𝑝 + 𝑞 ) + 𝑟 ) ↔ ∃ 𝑟 ∈ ℙ 7 = ( ( 2 + 𝑞 ) + 𝑟 ) ) ) |
12 |
|
oveq2 |
⊢ ( 𝑞 = 2 → ( 2 + 𝑞 ) = ( 2 + 2 ) ) |
13 |
12
|
oveq1d |
⊢ ( 𝑞 = 2 → ( ( 2 + 𝑞 ) + 𝑟 ) = ( ( 2 + 2 ) + 𝑟 ) ) |
14 |
13
|
eqeq2d |
⊢ ( 𝑞 = 2 → ( 7 = ( ( 2 + 𝑞 ) + 𝑟 ) ↔ 7 = ( ( 2 + 2 ) + 𝑟 ) ) ) |
15 |
14
|
rexbidv |
⊢ ( 𝑞 = 2 → ( ∃ 𝑟 ∈ ℙ 7 = ( ( 2 + 𝑞 ) + 𝑟 ) ↔ ∃ 𝑟 ∈ ℙ 7 = ( ( 2 + 2 ) + 𝑟 ) ) ) |
16 |
11 15
|
rspc2ev |
⊢ ( ( 2 ∈ ℙ ∧ 2 ∈ ℙ ∧ ∃ 𝑟 ∈ ℙ 7 = ( ( 2 + 2 ) + 𝑟 ) ) → ∃ 𝑝 ∈ ℙ ∃ 𝑞 ∈ ℙ ∃ 𝑟 ∈ ℙ 7 = ( ( 𝑝 + 𝑞 ) + 𝑟 ) ) |
17 |
2 2 7 16
|
mp3an |
⊢ ∃ 𝑝 ∈ ℙ ∃ 𝑞 ∈ ℙ ∃ 𝑟 ∈ ℙ 7 = ( ( 𝑝 + 𝑞 ) + 𝑟 ) |
18 |
|
isgbow |
⊢ ( 7 ∈ GoldbachOddW ↔ ( 7 ∈ Odd ∧ ∃ 𝑝 ∈ ℙ ∃ 𝑞 ∈ ℙ ∃ 𝑟 ∈ ℙ 7 = ( ( 𝑝 + 𝑞 ) + 𝑟 ) ) ) |
19 |
1 17 18
|
mpbir2an |
⊢ 7 ∈ GoldbachOddW |