Step |
Hyp |
Ref |
Expression |
1 |
|
6even |
⊢ 6 ∈ Even |
2 |
|
3prm |
⊢ 3 ∈ ℙ |
3 |
|
3odd |
⊢ 3 ∈ Odd |
4 |
|
gbpart6 |
⊢ 6 = ( 3 + 3 ) |
5 |
3 3 4
|
3pm3.2i |
⊢ ( 3 ∈ Odd ∧ 3 ∈ Odd ∧ 6 = ( 3 + 3 ) ) |
6 |
|
eleq1 |
⊢ ( 𝑝 = 3 → ( 𝑝 ∈ Odd ↔ 3 ∈ Odd ) ) |
7 |
|
biidd |
⊢ ( 𝑝 = 3 → ( 𝑞 ∈ Odd ↔ 𝑞 ∈ Odd ) ) |
8 |
|
oveq1 |
⊢ ( 𝑝 = 3 → ( 𝑝 + 𝑞 ) = ( 3 + 𝑞 ) ) |
9 |
8
|
eqeq2d |
⊢ ( 𝑝 = 3 → ( 6 = ( 𝑝 + 𝑞 ) ↔ 6 = ( 3 + 𝑞 ) ) ) |
10 |
6 7 9
|
3anbi123d |
⊢ ( 𝑝 = 3 → ( ( 𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 6 = ( 𝑝 + 𝑞 ) ) ↔ ( 3 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 6 = ( 3 + 𝑞 ) ) ) ) |
11 |
|
biidd |
⊢ ( 𝑞 = 3 → ( 3 ∈ Odd ↔ 3 ∈ Odd ) ) |
12 |
|
eleq1 |
⊢ ( 𝑞 = 3 → ( 𝑞 ∈ Odd ↔ 3 ∈ Odd ) ) |
13 |
|
oveq2 |
⊢ ( 𝑞 = 3 → ( 3 + 𝑞 ) = ( 3 + 3 ) ) |
14 |
13
|
eqeq2d |
⊢ ( 𝑞 = 3 → ( 6 = ( 3 + 𝑞 ) ↔ 6 = ( 3 + 3 ) ) ) |
15 |
11 12 14
|
3anbi123d |
⊢ ( 𝑞 = 3 → ( ( 3 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 6 = ( 3 + 𝑞 ) ) ↔ ( 3 ∈ Odd ∧ 3 ∈ Odd ∧ 6 = ( 3 + 3 ) ) ) ) |
16 |
10 15
|
rspc2ev |
⊢ ( ( 3 ∈ ℙ ∧ 3 ∈ ℙ ∧ ( 3 ∈ Odd ∧ 3 ∈ Odd ∧ 6 = ( 3 + 3 ) ) ) → ∃ 𝑝 ∈ ℙ ∃ 𝑞 ∈ ℙ ( 𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 6 = ( 𝑝 + 𝑞 ) ) ) |
17 |
2 2 5 16
|
mp3an |
⊢ ∃ 𝑝 ∈ ℙ ∃ 𝑞 ∈ ℙ ( 𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 6 = ( 𝑝 + 𝑞 ) ) |
18 |
|
isgbe |
⊢ ( 6 ∈ GoldbachEven ↔ ( 6 ∈ Even ∧ ∃ 𝑝 ∈ ℙ ∃ 𝑞 ∈ ℙ ( 𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 6 = ( 𝑝 + 𝑞 ) ) ) ) |
19 |
1 17 18
|
mpbir2an |
⊢ 6 ∈ GoldbachEven |