Step |
Hyp |
Ref |
Expression |
1 |
|
df-sb |
⊢ ( [ 𝑥 / 𝑥 ] 𝜑 ↔ ∀ 𝑦 ( 𝑦 = 𝑥 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) |
2 |
|
sp |
⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → ( 𝑥 = 𝑦 → 𝜑 ) ) |
3 |
2
|
imim2i |
⊢ ( ( 𝑦 = 𝑥 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) → ( 𝑦 = 𝑥 → ( 𝑥 = 𝑦 → 𝜑 ) ) ) |
4 |
3
|
alimi |
⊢ ( ∀ 𝑦 ( 𝑦 = 𝑥 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) → ∀ 𝑦 ( 𝑦 = 𝑥 → ( 𝑥 = 𝑦 → 𝜑 ) ) ) |
5 |
|
pm2.21 |
⊢ ( ¬ 𝑦 = 𝑥 → ( 𝑦 = 𝑥 → 𝜑 ) ) |
6 |
|
equcomi |
⊢ ( 𝑦 = 𝑥 → 𝑥 = 𝑦 ) |
7 |
6
|
imim1i |
⊢ ( ( 𝑥 = 𝑦 → 𝜑 ) → ( 𝑦 = 𝑥 → 𝜑 ) ) |
8 |
5 7
|
ja |
⊢ ( ( 𝑦 = 𝑥 → ( 𝑥 = 𝑦 → 𝜑 ) ) → ( 𝑦 = 𝑥 → 𝜑 ) ) |
9 |
8
|
alimi |
⊢ ( ∀ 𝑦 ( 𝑦 = 𝑥 → ( 𝑥 = 𝑦 → 𝜑 ) ) → ∀ 𝑦 ( 𝑦 = 𝑥 → 𝜑 ) ) |
10 |
|
ax6ev |
⊢ ∃ 𝑦 𝑦 = 𝑥 |
11 |
|
19.23v |
⊢ ( ∀ 𝑦 ( 𝑦 = 𝑥 → 𝜑 ) ↔ ( ∃ 𝑦 𝑦 = 𝑥 → 𝜑 ) ) |
12 |
11
|
biimpi |
⊢ ( ∀ 𝑦 ( 𝑦 = 𝑥 → 𝜑 ) → ( ∃ 𝑦 𝑦 = 𝑥 → 𝜑 ) ) |
13 |
9 10 12
|
mpisyl |
⊢ ( ∀ 𝑦 ( 𝑦 = 𝑥 → ( 𝑥 = 𝑦 → 𝜑 ) ) → 𝜑 ) |
14 |
4 13
|
syl |
⊢ ( ∀ 𝑦 ( 𝑦 = 𝑥 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) → 𝜑 ) |
15 |
1 14
|
sylbi |
⊢ ( [ 𝑥 / 𝑥 ] 𝜑 → 𝜑 ) |