Step |
Hyp |
Ref |
Expression |
1 |
|
sp |
⊢ ( ∀ 𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦 ) |
2 |
|
sbequ2 |
⊢ ( 𝑥 = 𝑦 → ( [ 𝑦 / 𝑥 ] 𝜑 → 𝜑 ) ) |
3 |
2
|
sps |
⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( [ 𝑦 / 𝑥 ] 𝜑 → 𝜑 ) ) |
4 |
|
orc |
⊢ ( ( 𝑥 = 𝑦 ∧ 𝜑 ) → ( ( 𝑥 = 𝑦 ∧ 𝜑 ) ∨ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) |
5 |
1 3 4
|
syl6an |
⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( [ 𝑦 / 𝑥 ] 𝜑 → ( ( 𝑥 = 𝑦 ∧ 𝜑 ) ∨ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) ) |
6 |
|
sb4b |
⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( [ 𝑦 / 𝑥 ] 𝜑 ↔ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) |
7 |
|
olc |
⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → ( ( 𝑥 = 𝑦 ∧ 𝜑 ) ∨ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) |
8 |
6 7
|
syl6bi |
⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( [ 𝑦 / 𝑥 ] 𝜑 → ( ( 𝑥 = 𝑦 ∧ 𝜑 ) ∨ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) ) |
9 |
5 8
|
pm2.61i |
⊢ ( [ 𝑦 / 𝑥 ] 𝜑 → ( ( 𝑥 = 𝑦 ∧ 𝜑 ) ∨ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) |
10 |
|
sbequ1 |
⊢ ( 𝑥 = 𝑦 → ( 𝜑 → [ 𝑦 / 𝑥 ] 𝜑 ) ) |
11 |
10
|
imp |
⊢ ( ( 𝑥 = 𝑦 ∧ 𝜑 ) → [ 𝑦 / 𝑥 ] 𝜑 ) |
12 |
|
sb2 |
⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → [ 𝑦 / 𝑥 ] 𝜑 ) |
13 |
11 12
|
jaoi |
⊢ ( ( ( 𝑥 = 𝑦 ∧ 𝜑 ) ∨ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) → [ 𝑦 / 𝑥 ] 𝜑 ) |
14 |
9 13
|
impbii |
⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ ( ( 𝑥 = 𝑦 ∧ 𝜑 ) ∨ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) |