Step |
Hyp |
Ref |
Expression |
1 |
|
nfna1 |
⊢ Ⅎ 𝑥 ¬ ∀ 𝑥 𝑥 = 𝑡 |
2 |
|
nfeqf2 |
⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑡 → Ⅎ 𝑥 𝑦 = 𝑡 ) |
3 |
1 2
|
nfan1 |
⊢ Ⅎ 𝑥 ( ¬ ∀ 𝑥 𝑥 = 𝑡 ∧ 𝑦 = 𝑡 ) |
4 |
|
equequ2 |
⊢ ( 𝑦 = 𝑡 → ( 𝑥 = 𝑦 ↔ 𝑥 = 𝑡 ) ) |
5 |
4
|
imbi1d |
⊢ ( 𝑦 = 𝑡 → ( ( 𝑥 = 𝑦 → 𝜑 ) ↔ ( 𝑥 = 𝑡 → 𝜑 ) ) ) |
6 |
5
|
adantl |
⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑡 ∧ 𝑦 = 𝑡 ) → ( ( 𝑥 = 𝑦 → 𝜑 ) ↔ ( 𝑥 = 𝑡 → 𝜑 ) ) ) |
7 |
3 6
|
albid |
⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑡 ∧ 𝑦 = 𝑡 ) → ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) ) |
8 |
7
|
pm5.74da |
⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑡 → ( ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ↔ ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) ) ) |
9 |
8
|
albidv |
⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑡 → ( ∀ 𝑦 ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ↔ ∀ 𝑦 ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) ) ) |
10 |
|
df-sb |
⊢ ( [ 𝑡 / 𝑥 ] 𝜑 ↔ ∀ 𝑦 ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) |
11 |
|
ax6ev |
⊢ ∃ 𝑦 𝑦 = 𝑡 |
12 |
11
|
a1bi |
⊢ ( ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ↔ ( ∃ 𝑦 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) ) |
13 |
|
19.23v |
⊢ ( ∀ 𝑦 ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) ↔ ( ∃ 𝑦 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) ) |
14 |
12 13
|
bitr4i |
⊢ ( ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ↔ ∀ 𝑦 ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) ) |
15 |
9 10 14
|
3bitr4g |
⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑡 → ( [ 𝑡 / 𝑥 ] 𝜑 ↔ ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) ) |