Step |
Hyp |
Ref |
Expression |
1 |
|
biimt |
⊢ ( 𝐴 ∈ 𝐵 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ ( 𝐴 ∈ 𝐵 → ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ) ) ) |
2 |
|
df-ral |
⊢ ( ∀ 𝑥 ∈ 𝐵 ( 𝑥 = 𝐴 → 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐵 → ( 𝑥 = 𝐴 → 𝜑 ) ) ) |
3 |
|
eleq1 |
⊢ ( 𝑥 = 𝐴 → ( 𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵 ) ) |
4 |
3
|
pm5.32ri |
⊢ ( ( 𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴 ) ↔ ( 𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴 ) ) |
5 |
4
|
imbi1i |
⊢ ( ( ( 𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴 ) → 𝜑 ) ↔ ( ( 𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴 ) → 𝜑 ) ) |
6 |
|
impexp |
⊢ ( ( ( 𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴 ) → 𝜑 ) ↔ ( 𝑥 ∈ 𝐵 → ( 𝑥 = 𝐴 → 𝜑 ) ) ) |
7 |
|
impexp |
⊢ ( ( ( 𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴 ) → 𝜑 ) ↔ ( 𝐴 ∈ 𝐵 → ( 𝑥 = 𝐴 → 𝜑 ) ) ) |
8 |
5 6 7
|
3bitr3i |
⊢ ( ( 𝑥 ∈ 𝐵 → ( 𝑥 = 𝐴 → 𝜑 ) ) ↔ ( 𝐴 ∈ 𝐵 → ( 𝑥 = 𝐴 → 𝜑 ) ) ) |
9 |
8
|
albii |
⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐵 → ( 𝑥 = 𝐴 → 𝜑 ) ) ↔ ∀ 𝑥 ( 𝐴 ∈ 𝐵 → ( 𝑥 = 𝐴 → 𝜑 ) ) ) |
10 |
|
19.21v |
⊢ ( ∀ 𝑥 ( 𝐴 ∈ 𝐵 → ( 𝑥 = 𝐴 → 𝜑 ) ) ↔ ( 𝐴 ∈ 𝐵 → ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ) ) |
11 |
2 9 10
|
3bitrri |
⊢ ( ( 𝐴 ∈ 𝐵 → ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ) ↔ ∀ 𝑥 ∈ 𝐵 ( 𝑥 = 𝐴 → 𝜑 ) ) |
12 |
1 11
|
bitrdi |
⊢ ( 𝐴 ∈ 𝐵 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ ∀ 𝑥 ∈ 𝐵 ( 𝑥 = 𝐴 → 𝜑 ) ) ) |
13 |
12
|
3ad2ant3 |
⊢ ( ( Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) ∧ 𝐴 ∈ 𝐵 ) → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ ∀ 𝑥 ∈ 𝐵 ( 𝑥 = 𝐴 → 𝜑 ) ) ) |
14 |
|
ceqsalt |
⊢ ( ( Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) ∧ 𝐴 ∈ 𝐵 ) → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ 𝜓 ) ) |
15 |
13 14
|
bitr3d |
⊢ ( ( Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) ∧ 𝐴 ∈ 𝐵 ) → ( ∀ 𝑥 ∈ 𝐵 ( 𝑥 = 𝐴 → 𝜑 ) ↔ 𝜓 ) ) |