Step |
Hyp |
Ref |
Expression |
1 |
|
breq1 |
⊢ ( 𝐴 = 𝐵 → ( 𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥 ) ) |
2 |
1
|
ralrimivw |
⊢ ( 𝐴 = 𝐵 → ∀ 𝑥 ∈ ℕ0 ( 𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥 ) ) |
3 |
|
simpll |
⊢ ( ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ∧ ∀ 𝑥 ∈ ℕ0 ( 𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥 ) ) → 𝐴 ∈ ℕ0 ) |
4 |
|
simplr |
⊢ ( ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ∧ ∀ 𝑥 ∈ ℕ0 ( 𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥 ) ) → 𝐵 ∈ ℕ0 ) |
5 |
|
nn0z |
⊢ ( 𝐵 ∈ ℕ0 → 𝐵 ∈ ℤ ) |
6 |
|
iddvds |
⊢ ( 𝐵 ∈ ℤ → 𝐵 ∥ 𝐵 ) |
7 |
5 6
|
syl |
⊢ ( 𝐵 ∈ ℕ0 → 𝐵 ∥ 𝐵 ) |
8 |
7
|
ad2antlr |
⊢ ( ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ∧ ∀ 𝑥 ∈ ℕ0 ( 𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥 ) ) → 𝐵 ∥ 𝐵 ) |
9 |
|
breq2 |
⊢ ( 𝑥 = 𝐵 → ( 𝐴 ∥ 𝑥 ↔ 𝐴 ∥ 𝐵 ) ) |
10 |
|
breq2 |
⊢ ( 𝑥 = 𝐵 → ( 𝐵 ∥ 𝑥 ↔ 𝐵 ∥ 𝐵 ) ) |
11 |
9 10
|
bibi12d |
⊢ ( 𝑥 = 𝐵 → ( ( 𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥 ) ↔ ( 𝐴 ∥ 𝐵 ↔ 𝐵 ∥ 𝐵 ) ) ) |
12 |
11
|
rspcva |
⊢ ( ( 𝐵 ∈ ℕ0 ∧ ∀ 𝑥 ∈ ℕ0 ( 𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥 ) ) → ( 𝐴 ∥ 𝐵 ↔ 𝐵 ∥ 𝐵 ) ) |
13 |
12
|
adantll |
⊢ ( ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ∧ ∀ 𝑥 ∈ ℕ0 ( 𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥 ) ) → ( 𝐴 ∥ 𝐵 ↔ 𝐵 ∥ 𝐵 ) ) |
14 |
8 13
|
mpbird |
⊢ ( ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ∧ ∀ 𝑥 ∈ ℕ0 ( 𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥 ) ) → 𝐴 ∥ 𝐵 ) |
15 |
|
nn0z |
⊢ ( 𝐴 ∈ ℕ0 → 𝐴 ∈ ℤ ) |
16 |
|
iddvds |
⊢ ( 𝐴 ∈ ℤ → 𝐴 ∥ 𝐴 ) |
17 |
15 16
|
syl |
⊢ ( 𝐴 ∈ ℕ0 → 𝐴 ∥ 𝐴 ) |
18 |
17
|
ad2antrr |
⊢ ( ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ∧ ∀ 𝑥 ∈ ℕ0 ( 𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥 ) ) → 𝐴 ∥ 𝐴 ) |
19 |
|
breq2 |
⊢ ( 𝑥 = 𝐴 → ( 𝐴 ∥ 𝑥 ↔ 𝐴 ∥ 𝐴 ) ) |
20 |
|
breq2 |
⊢ ( 𝑥 = 𝐴 → ( 𝐵 ∥ 𝑥 ↔ 𝐵 ∥ 𝐴 ) ) |
21 |
19 20
|
bibi12d |
⊢ ( 𝑥 = 𝐴 → ( ( 𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥 ) ↔ ( 𝐴 ∥ 𝐴 ↔ 𝐵 ∥ 𝐴 ) ) ) |
22 |
21
|
rspcva |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ ∀ 𝑥 ∈ ℕ0 ( 𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥 ) ) → ( 𝐴 ∥ 𝐴 ↔ 𝐵 ∥ 𝐴 ) ) |
23 |
22
|
adantlr |
⊢ ( ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ∧ ∀ 𝑥 ∈ ℕ0 ( 𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥 ) ) → ( 𝐴 ∥ 𝐴 ↔ 𝐵 ∥ 𝐴 ) ) |
24 |
18 23
|
mpbid |
⊢ ( ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ∧ ∀ 𝑥 ∈ ℕ0 ( 𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥 ) ) → 𝐵 ∥ 𝐴 ) |
25 |
|
dvdseq |
⊢ ( ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ∧ ( 𝐴 ∥ 𝐵 ∧ 𝐵 ∥ 𝐴 ) ) → 𝐴 = 𝐵 ) |
26 |
3 4 14 24 25
|
syl22anc |
⊢ ( ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ∧ ∀ 𝑥 ∈ ℕ0 ( 𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥 ) ) → 𝐴 = 𝐵 ) |
27 |
26
|
ex |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) → ( ∀ 𝑥 ∈ ℕ0 ( 𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥 ) → 𝐴 = 𝐵 ) ) |
28 |
2 27
|
impbid2 |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) → ( 𝐴 = 𝐵 ↔ ∀ 𝑥 ∈ ℕ0 ( 𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥 ) ) ) |