Step |
Hyp |
Ref |
Expression |
1 |
|
orcom |
⊢ ( ( 𝑥 = 𝑦 ∨ ( 𝐶 ∩ 𝐷 ) = ∅ ) ↔ ( ( 𝐶 ∩ 𝐷 ) = ∅ ∨ 𝑥 = 𝑦 ) ) |
2 |
|
df-or |
⊢ ( ( ( 𝐶 ∩ 𝐷 ) = ∅ ∨ 𝑥 = 𝑦 ) ↔ ( ¬ ( 𝐶 ∩ 𝐷 ) = ∅ → 𝑥 = 𝑦 ) ) |
3 |
|
neq0 |
⊢ ( ¬ ( 𝐶 ∩ 𝐷 ) = ∅ ↔ ∃ 𝑧 𝑧 ∈ ( 𝐶 ∩ 𝐷 ) ) |
4 |
|
elin |
⊢ ( 𝑧 ∈ ( 𝐶 ∩ 𝐷 ) ↔ ( 𝑧 ∈ 𝐶 ∧ 𝑧 ∈ 𝐷 ) ) |
5 |
4
|
exbii |
⊢ ( ∃ 𝑧 𝑧 ∈ ( 𝐶 ∩ 𝐷 ) ↔ ∃ 𝑧 ( 𝑧 ∈ 𝐶 ∧ 𝑧 ∈ 𝐷 ) ) |
6 |
3 5
|
bitri |
⊢ ( ¬ ( 𝐶 ∩ 𝐷 ) = ∅ ↔ ∃ 𝑧 ( 𝑧 ∈ 𝐶 ∧ 𝑧 ∈ 𝐷 ) ) |
7 |
6
|
imbi1i |
⊢ ( ( ¬ ( 𝐶 ∩ 𝐷 ) = ∅ → 𝑥 = 𝑦 ) ↔ ( ∃ 𝑧 ( 𝑧 ∈ 𝐶 ∧ 𝑧 ∈ 𝐷 ) → 𝑥 = 𝑦 ) ) |
8 |
|
19.23v |
⊢ ( ∀ 𝑧 ( ( 𝑧 ∈ 𝐶 ∧ 𝑧 ∈ 𝐷 ) → 𝑥 = 𝑦 ) ↔ ( ∃ 𝑧 ( 𝑧 ∈ 𝐶 ∧ 𝑧 ∈ 𝐷 ) → 𝑥 = 𝑦 ) ) |
9 |
7 8
|
bitr4i |
⊢ ( ( ¬ ( 𝐶 ∩ 𝐷 ) = ∅ → 𝑥 = 𝑦 ) ↔ ∀ 𝑧 ( ( 𝑧 ∈ 𝐶 ∧ 𝑧 ∈ 𝐷 ) → 𝑥 = 𝑦 ) ) |
10 |
1 2 9
|
3bitri |
⊢ ( ( 𝑥 = 𝑦 ∨ ( 𝐶 ∩ 𝐷 ) = ∅ ) ↔ ∀ 𝑧 ( ( 𝑧 ∈ 𝐶 ∧ 𝑧 ∈ 𝐷 ) → 𝑥 = 𝑦 ) ) |
11 |
10
|
ralbii |
⊢ ( ∀ 𝑦 ∈ 𝐵 ( 𝑥 = 𝑦 ∨ ( 𝐶 ∩ 𝐷 ) = ∅ ) ↔ ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ( ( 𝑧 ∈ 𝐶 ∧ 𝑧 ∈ 𝐷 ) → 𝑥 = 𝑦 ) ) |
12 |
|
ralcom4 |
⊢ ( ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ( ( 𝑧 ∈ 𝐶 ∧ 𝑧 ∈ 𝐷 ) → 𝑥 = 𝑦 ) ↔ ∀ 𝑧 ∀ 𝑦 ∈ 𝐵 ( ( 𝑧 ∈ 𝐶 ∧ 𝑧 ∈ 𝐷 ) → 𝑥 = 𝑦 ) ) |
13 |
11 12
|
bitri |
⊢ ( ∀ 𝑦 ∈ 𝐵 ( 𝑥 = 𝑦 ∨ ( 𝐶 ∩ 𝐷 ) = ∅ ) ↔ ∀ 𝑧 ∀ 𝑦 ∈ 𝐵 ( ( 𝑧 ∈ 𝐶 ∧ 𝑧 ∈ 𝐷 ) → 𝑥 = 𝑦 ) ) |
14 |
13
|
ralbii |
⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝑥 = 𝑦 ∨ ( 𝐶 ∩ 𝐷 ) = ∅ ) ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑧 ∀ 𝑦 ∈ 𝐵 ( ( 𝑧 ∈ 𝐶 ∧ 𝑧 ∈ 𝐷 ) → 𝑥 = 𝑦 ) ) |