Step |
Hyp |
Ref |
Expression |
1 |
|
sb8f.nf |
⊢ Ⅎ 𝑦 𝜑 |
2 |
|
sb6 |
⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) |
3 |
2
|
albii |
⊢ ( ∀ 𝑦 [ 𝑦 / 𝑥 ] 𝜑 ↔ ∀ 𝑦 ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) |
4 |
|
alcom |
⊢ ( ∀ 𝑦 ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ↔ ∀ 𝑥 ∀ 𝑦 ( 𝑥 = 𝑦 → 𝜑 ) ) |
5 |
|
sb6 |
⊢ ( [ 𝑥 / 𝑦 ] 𝜑 ↔ ∀ 𝑦 ( 𝑦 = 𝑥 → 𝜑 ) ) |
6 |
1
|
sbf |
⊢ ( [ 𝑥 / 𝑦 ] 𝜑 ↔ 𝜑 ) |
7 |
|
equcom |
⊢ ( 𝑦 = 𝑥 ↔ 𝑥 = 𝑦 ) |
8 |
7
|
imbi1i |
⊢ ( ( 𝑦 = 𝑥 → 𝜑 ) ↔ ( 𝑥 = 𝑦 → 𝜑 ) ) |
9 |
8
|
albii |
⊢ ( ∀ 𝑦 ( 𝑦 = 𝑥 → 𝜑 ) ↔ ∀ 𝑦 ( 𝑥 = 𝑦 → 𝜑 ) ) |
10 |
5 6 9
|
3bitr3ri |
⊢ ( ∀ 𝑦 ( 𝑥 = 𝑦 → 𝜑 ) ↔ 𝜑 ) |
11 |
10
|
albii |
⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝑥 = 𝑦 → 𝜑 ) ↔ ∀ 𝑥 𝜑 ) |
12 |
3 4 11
|
3bitrri |
⊢ ( ∀ 𝑥 𝜑 ↔ ∀ 𝑦 [ 𝑦 / 𝑥 ] 𝜑 ) |