Step |
Hyp |
Ref |
Expression |
1 |
|
sb6 |
⊢ ( [ 𝑦 / 𝑥 ] ( 𝜑 → 𝜓 ) ↔ ∀ 𝑥 ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) ) ) |
2 |
|
bi2.04 |
⊢ ( ( 𝜑 → ( 𝑥 = 𝑦 → 𝜓 ) ) ↔ ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) ) ) |
3 |
2
|
albii |
⊢ ( ∀ 𝑥 ( 𝜑 → ( 𝑥 = 𝑦 → 𝜓 ) ) ↔ ∀ 𝑥 ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) ) ) |
4 |
|
19.21v |
⊢ ( ∀ 𝑥 ( 𝜑 → ( 𝑥 = 𝑦 → 𝜓 ) ) ↔ ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜓 ) ) ) |
5 |
1 3 4
|
3bitr2i |
⊢ ( [ 𝑦 / 𝑥 ] ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜓 ) ) ) |
6 |
|
sb6 |
⊢ ( [ 𝑦 / 𝑥 ] 𝜓 ↔ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜓 ) ) |
7 |
6
|
imbi2i |
⊢ ( ( 𝜑 → [ 𝑦 / 𝑥 ] 𝜓 ) ↔ ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜓 ) ) ) |
8 |
5 7
|
bitr4i |
⊢ ( [ 𝑦 / 𝑥 ] ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → [ 𝑦 / 𝑥 ] 𝜓 ) ) |