Step |
Hyp |
Ref |
Expression |
1 |
|
8even |
⊢ 8 ∈ Even |
2 |
|
5prm |
⊢ 5 ∈ ℙ |
3 |
|
3prm |
⊢ 3 ∈ ℙ |
4 |
|
5odd |
⊢ 5 ∈ Odd |
5 |
|
3odd |
⊢ 3 ∈ Odd |
6 |
|
5p3e8 |
⊢ ( 5 + 3 ) = 8 |
7 |
6
|
eqcomi |
⊢ 8 = ( 5 + 3 ) |
8 |
4 5 7
|
3pm3.2i |
⊢ ( 5 ∈ Odd ∧ 3 ∈ Odd ∧ 8 = ( 5 + 3 ) ) |
9 |
|
eleq1 |
⊢ ( 𝑝 = 5 → ( 𝑝 ∈ Odd ↔ 5 ∈ Odd ) ) |
10 |
|
biidd |
⊢ ( 𝑝 = 5 → ( 𝑞 ∈ Odd ↔ 𝑞 ∈ 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 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 8 = ( 𝑝 + 𝑞 ) ) ↔ ( 5 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 8 = ( 5 + 𝑞 ) ) ) ) |
14 |
|
biidd |
⊢ ( 𝑞 = 3 → ( 5 ∈ Odd ↔ 5 ∈ Odd ) ) |
15 |
|
eleq1 |
⊢ ( 𝑞 = 3 → ( 𝑞 ∈ Odd ↔ 3 ∈ 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 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 8 = ( 5 + 𝑞 ) ) ↔ ( 5 ∈ Odd ∧ 3 ∈ Odd ∧ 8 = ( 5 + 3 ) ) ) ) |
19 |
13 18
|
rspc2ev |
⊢ ( ( 5 ∈ ℙ ∧ 3 ∈ ℙ ∧ ( 5 ∈ Odd ∧ 3 ∈ Odd ∧ 8 = ( 5 + 3 ) ) ) → ∃ 𝑝 ∈ ℙ ∃ 𝑞 ∈ ℙ ( 𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 8 = ( 𝑝 + 𝑞 ) ) ) |
20 |
2 3 8 19
|
mp3an |
⊢ ∃ 𝑝 ∈ ℙ ∃ 𝑞 ∈ ℙ ( 𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 8 = ( 𝑝 + 𝑞 ) ) |
21 |
|
isgbe |
⊢ ( 8 ∈ GoldbachEven ↔ ( 8 ∈ Even ∧ ∃ 𝑝 ∈ ℙ ∃ 𝑞 ∈ ℙ ( 𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 8 = ( 𝑝 + 𝑞 ) ) ) ) |
22 |
1 20 21
|
mpbir2an |
⊢ 8 ∈ GoldbachEven |