Step |
Hyp |
Ref |
Expression |
1 |
|
cbvdisjf.1 |
⊢ Ⅎ 𝑥 𝐴 |
2 |
|
cbvdisjf.2 |
⊢ Ⅎ 𝑦 𝐵 |
3 |
|
cbvdisjf.3 |
⊢ Ⅎ 𝑥 𝐶 |
4 |
|
cbvdisjf.4 |
⊢ ( 𝑥 = 𝑦 → 𝐵 = 𝐶 ) |
5 |
|
nfv |
⊢ Ⅎ 𝑦 𝑥 ∈ 𝐴 |
6 |
2
|
nfcri |
⊢ Ⅎ 𝑦 𝑧 ∈ 𝐵 |
7 |
5 6
|
nfan |
⊢ Ⅎ 𝑦 ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) |
8 |
1
|
nfcri |
⊢ Ⅎ 𝑥 𝑦 ∈ 𝐴 |
9 |
3
|
nfcri |
⊢ Ⅎ 𝑥 𝑧 ∈ 𝐶 |
10 |
8 9
|
nfan |
⊢ Ⅎ 𝑥 ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶 ) |
11 |
|
eleq1w |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴 ) ) |
12 |
4
|
eleq2d |
⊢ ( 𝑥 = 𝑦 → ( 𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶 ) ) |
13 |
11 12
|
anbi12d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) ↔ ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶 ) ) ) |
14 |
7 10 13
|
cbvmow |
⊢ ( ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) ↔ ∃* 𝑦 ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶 ) ) |
15 |
|
df-rmo |
⊢ ( ∃* 𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) ) |
16 |
|
df-rmo |
⊢ ( ∃* 𝑦 ∈ 𝐴 𝑧 ∈ 𝐶 ↔ ∃* 𝑦 ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶 ) ) |
17 |
14 15 16
|
3bitr4i |
⊢ ( ∃* 𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃* 𝑦 ∈ 𝐴 𝑧 ∈ 𝐶 ) |
18 |
17
|
albii |
⊢ ( ∀ 𝑧 ∃* 𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∀ 𝑧 ∃* 𝑦 ∈ 𝐴 𝑧 ∈ 𝐶 ) |
19 |
|
df-disj |
⊢ ( Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀ 𝑧 ∃* 𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ) |
20 |
|
df-disj |
⊢ ( Disj 𝑦 ∈ 𝐴 𝐶 ↔ ∀ 𝑧 ∃* 𝑦 ∈ 𝐴 𝑧 ∈ 𝐶 ) |
21 |
18 19 20
|
3bitr4i |
⊢ ( Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑦 ∈ 𝐴 𝐶 ) |