Step |
Hyp |
Ref |
Expression |
1 |
|
alcom |
⊢ ( ∀ 𝑤 ∀ 𝑥 ( 𝑤 = 𝑧 → ∀ 𝑦 ( 𝑦 = 𝑤 → 𝜑 ) ) ↔ ∀ 𝑥 ∀ 𝑤 ( 𝑤 = 𝑧 → ∀ 𝑦 ( 𝑦 = 𝑤 → 𝜑 ) ) ) |
2 |
|
19.21v |
⊢ ( ∀ 𝑥 ( 𝑦 = 𝑤 → 𝜑 ) ↔ ( 𝑦 = 𝑤 → ∀ 𝑥 𝜑 ) ) |
3 |
2
|
albii |
⊢ ( ∀ 𝑦 ∀ 𝑥 ( 𝑦 = 𝑤 → 𝜑 ) ↔ ∀ 𝑦 ( 𝑦 = 𝑤 → ∀ 𝑥 𝜑 ) ) |
4 |
|
alcom |
⊢ ( ∀ 𝑦 ∀ 𝑥 ( 𝑦 = 𝑤 → 𝜑 ) ↔ ∀ 𝑥 ∀ 𝑦 ( 𝑦 = 𝑤 → 𝜑 ) ) |
5 |
3 4
|
bitr3i |
⊢ ( ∀ 𝑦 ( 𝑦 = 𝑤 → ∀ 𝑥 𝜑 ) ↔ ∀ 𝑥 ∀ 𝑦 ( 𝑦 = 𝑤 → 𝜑 ) ) |
6 |
5
|
imbi2i |
⊢ ( ( 𝑤 = 𝑧 → ∀ 𝑦 ( 𝑦 = 𝑤 → ∀ 𝑥 𝜑 ) ) ↔ ( 𝑤 = 𝑧 → ∀ 𝑥 ∀ 𝑦 ( 𝑦 = 𝑤 → 𝜑 ) ) ) |
7 |
6
|
albii |
⊢ ( ∀ 𝑤 ( 𝑤 = 𝑧 → ∀ 𝑦 ( 𝑦 = 𝑤 → ∀ 𝑥 𝜑 ) ) ↔ ∀ 𝑤 ( 𝑤 = 𝑧 → ∀ 𝑥 ∀ 𝑦 ( 𝑦 = 𝑤 → 𝜑 ) ) ) |
8 |
|
df-sb |
⊢ ( [ 𝑧 / 𝑦 ] ∀ 𝑥 𝜑 ↔ ∀ 𝑤 ( 𝑤 = 𝑧 → ∀ 𝑦 ( 𝑦 = 𝑤 → ∀ 𝑥 𝜑 ) ) ) |
9 |
|
19.21v |
⊢ ( ∀ 𝑥 ( 𝑤 = 𝑧 → ∀ 𝑦 ( 𝑦 = 𝑤 → 𝜑 ) ) ↔ ( 𝑤 = 𝑧 → ∀ 𝑥 ∀ 𝑦 ( 𝑦 = 𝑤 → 𝜑 ) ) ) |
10 |
9
|
albii |
⊢ ( ∀ 𝑤 ∀ 𝑥 ( 𝑤 = 𝑧 → ∀ 𝑦 ( 𝑦 = 𝑤 → 𝜑 ) ) ↔ ∀ 𝑤 ( 𝑤 = 𝑧 → ∀ 𝑥 ∀ 𝑦 ( 𝑦 = 𝑤 → 𝜑 ) ) ) |
11 |
7 8 10
|
3bitr4i |
⊢ ( [ 𝑧 / 𝑦 ] ∀ 𝑥 𝜑 ↔ ∀ 𝑤 ∀ 𝑥 ( 𝑤 = 𝑧 → ∀ 𝑦 ( 𝑦 = 𝑤 → 𝜑 ) ) ) |
12 |
|
df-sb |
⊢ ( [ 𝑧 / 𝑦 ] 𝜑 ↔ ∀ 𝑤 ( 𝑤 = 𝑧 → ∀ 𝑦 ( 𝑦 = 𝑤 → 𝜑 ) ) ) |
13 |
12
|
albii |
⊢ ( ∀ 𝑥 [ 𝑧 / 𝑦 ] 𝜑 ↔ ∀ 𝑥 ∀ 𝑤 ( 𝑤 = 𝑧 → ∀ 𝑦 ( 𝑦 = 𝑤 → 𝜑 ) ) ) |
14 |
1 11 13
|
3bitr4i |
⊢ ( [ 𝑧 / 𝑦 ] ∀ 𝑥 𝜑 ↔ ∀ 𝑥 [ 𝑧 / 𝑦 ] 𝜑 ) |