Step |
Hyp |
Ref |
Expression |
1 |
|
r19.23v |
⊢ ( ∀ 𝑦 ∈ 𝐴 ( 𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥 ) ↔ ( ∃ 𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥 ) ) |
2 |
1
|
albii |
⊢ ( ∀ 𝑥 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥 ) ↔ ∀ 𝑥 ( ∃ 𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥 ) ) |
3 |
|
ralcom4 |
⊢ ( ∀ 𝑦 ∈ 𝐴 ∀ 𝑥 ( 𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥 ) ↔ ∀ 𝑥 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥 ) ) |
4 |
|
eluni2 |
⊢ ( 𝑥 ∈ ∪ 𝐴 ↔ ∃ 𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ) |
5 |
4
|
imbi1i |
⊢ ( ( 𝑥 ∈ ∪ 𝐴 → ¬ 𝑥 ∈ 𝑥 ) ↔ ( ∃ 𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥 ) ) |
6 |
5
|
albii |
⊢ ( ∀ 𝑥 ( 𝑥 ∈ ∪ 𝐴 → ¬ 𝑥 ∈ 𝑥 ) ↔ ∀ 𝑥 ( ∃ 𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥 ) ) |
7 |
2 3 6
|
3bitr4ri |
⊢ ( ∀ 𝑥 ( 𝑥 ∈ ∪ 𝐴 → ¬ 𝑥 ∈ 𝑥 ) ↔ ∀ 𝑦 ∈ 𝐴 ∀ 𝑥 ( 𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥 ) ) |
8 |
|
df-ral |
⊢ ( ∀ 𝑥 ∈ ∪ 𝐴 ¬ 𝑥 ∈ 𝑥 ↔ ∀ 𝑥 ( 𝑥 ∈ ∪ 𝐴 → ¬ 𝑥 ∈ 𝑥 ) ) |
9 |
|
df-ral |
⊢ ( ∀ 𝑥 ∈ 𝑦 ¬ 𝑥 ∈ 𝑥 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥 ) ) |
10 |
9
|
ralbii |
⊢ ( ∀ 𝑦 ∈ 𝐴 ∀ 𝑥 ∈ 𝑦 ¬ 𝑥 ∈ 𝑥 ↔ ∀ 𝑦 ∈ 𝐴 ∀ 𝑥 ( 𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥 ) ) |
11 |
7 8 10
|
3bitr4i |
⊢ ( ∀ 𝑥 ∈ ∪ 𝐴 ¬ 𝑥 ∈ 𝑥 ↔ ∀ 𝑦 ∈ 𝐴 ∀ 𝑥 ∈ 𝑦 ¬ 𝑥 ∈ 𝑥 ) |