Step |
Hyp |
Ref |
Expression |
1 |
|
ndisj2.1 |
⊢ ( 𝑥 = 𝑦 → 𝐵 = 𝐶 ) |
2 |
1
|
disjor |
⊢ ( Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 = 𝑦 ∨ ( 𝐵 ∩ 𝐶 ) = ∅ ) ) |
3 |
2
|
notbii |
⊢ ( ¬ Disj 𝑥 ∈ 𝐴 𝐵 ↔ ¬ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 = 𝑦 ∨ ( 𝐵 ∩ 𝐶 ) = ∅ ) ) |
4 |
|
rexnal |
⊢ ( ∃ 𝑥 ∈ 𝐴 ¬ ∀ 𝑦 ∈ 𝐴 ( 𝑥 = 𝑦 ∨ ( 𝐵 ∩ 𝐶 ) = ∅ ) ↔ ¬ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 = 𝑦 ∨ ( 𝐵 ∩ 𝐶 ) = ∅ ) ) |
5 |
|
rexnal |
⊢ ( ∃ 𝑦 ∈ 𝐴 ¬ ( 𝑥 = 𝑦 ∨ ( 𝐵 ∩ 𝐶 ) = ∅ ) ↔ ¬ ∀ 𝑦 ∈ 𝐴 ( 𝑥 = 𝑦 ∨ ( 𝐵 ∩ 𝐶 ) = ∅ ) ) |
6 |
|
ioran |
⊢ ( ¬ ( 𝑥 = 𝑦 ∨ ( 𝐵 ∩ 𝐶 ) = ∅ ) ↔ ( ¬ 𝑥 = 𝑦 ∧ ¬ ( 𝐵 ∩ 𝐶 ) = ∅ ) ) |
7 |
|
df-ne |
⊢ ( 𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦 ) |
8 |
|
df-ne |
⊢ ( ( 𝐵 ∩ 𝐶 ) ≠ ∅ ↔ ¬ ( 𝐵 ∩ 𝐶 ) = ∅ ) |
9 |
7 8
|
anbi12i |
⊢ ( ( 𝑥 ≠ 𝑦 ∧ ( 𝐵 ∩ 𝐶 ) ≠ ∅ ) ↔ ( ¬ 𝑥 = 𝑦 ∧ ¬ ( 𝐵 ∩ 𝐶 ) = ∅ ) ) |
10 |
6 9
|
bitr4i |
⊢ ( ¬ ( 𝑥 = 𝑦 ∨ ( 𝐵 ∩ 𝐶 ) = ∅ ) ↔ ( 𝑥 ≠ 𝑦 ∧ ( 𝐵 ∩ 𝐶 ) ≠ ∅ ) ) |
11 |
10
|
rexbii |
⊢ ( ∃ 𝑦 ∈ 𝐴 ¬ ( 𝑥 = 𝑦 ∨ ( 𝐵 ∩ 𝐶 ) = ∅ ) ↔ ∃ 𝑦 ∈ 𝐴 ( 𝑥 ≠ 𝑦 ∧ ( 𝐵 ∩ 𝐶 ) ≠ ∅ ) ) |
12 |
5 11
|
bitr3i |
⊢ ( ¬ ∀ 𝑦 ∈ 𝐴 ( 𝑥 = 𝑦 ∨ ( 𝐵 ∩ 𝐶 ) = ∅ ) ↔ ∃ 𝑦 ∈ 𝐴 ( 𝑥 ≠ 𝑦 ∧ ( 𝐵 ∩ 𝐶 ) ≠ ∅ ) ) |
13 |
12
|
rexbii |
⊢ ( ∃ 𝑥 ∈ 𝐴 ¬ ∀ 𝑦 ∈ 𝐴 ( 𝑥 = 𝑦 ∨ ( 𝐵 ∩ 𝐶 ) = ∅ ) ↔ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐴 ( 𝑥 ≠ 𝑦 ∧ ( 𝐵 ∩ 𝐶 ) ≠ ∅ ) ) |
14 |
3 4 13
|
3bitr2i |
⊢ ( ¬ Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐴 ( 𝑥 ≠ 𝑦 ∧ ( 𝐵 ∩ 𝐶 ) ≠ ∅ ) ) |