Step |
Hyp |
Ref |
Expression |
1 |
|
sb4b |
⊢ ( ¬ ∀ 𝑦 𝑦 = 𝑧 → ( [ 𝑧 / 𝑦 ] ∀ 𝑥 𝜑 ↔ ∀ 𝑦 ( 𝑦 = 𝑧 → ∀ 𝑥 𝜑 ) ) ) |
2 |
|
nfnae |
⊢ Ⅎ 𝑦 ¬ ∀ 𝑥 𝑥 = 𝑦 |
3 |
|
nfnae |
⊢ Ⅎ 𝑦 ¬ ∀ 𝑥 𝑥 = 𝑧 |
4 |
2 3
|
nfan |
⊢ Ⅎ 𝑦 ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) |
5 |
|
nfeqf |
⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) → Ⅎ 𝑥 𝑦 = 𝑧 ) |
6 |
|
19.21t |
⊢ ( Ⅎ 𝑥 𝑦 = 𝑧 → ( ∀ 𝑥 ( 𝑦 = 𝑧 → 𝜑 ) ↔ ( 𝑦 = 𝑧 → ∀ 𝑥 𝜑 ) ) ) |
7 |
6
|
bicomd |
⊢ ( Ⅎ 𝑥 𝑦 = 𝑧 → ( ( 𝑦 = 𝑧 → ∀ 𝑥 𝜑 ) ↔ ∀ 𝑥 ( 𝑦 = 𝑧 → 𝜑 ) ) ) |
8 |
5 7
|
syl |
⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) → ( ( 𝑦 = 𝑧 → ∀ 𝑥 𝜑 ) ↔ ∀ 𝑥 ( 𝑦 = 𝑧 → 𝜑 ) ) ) |
9 |
4 8
|
albid |
⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) → ( ∀ 𝑦 ( 𝑦 = 𝑧 → ∀ 𝑥 𝜑 ) ↔ ∀ 𝑦 ∀ 𝑥 ( 𝑦 = 𝑧 → 𝜑 ) ) ) |
10 |
1 9
|
sylan9bbr |
⊢ ( ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) ∧ ¬ ∀ 𝑦 𝑦 = 𝑧 ) → ( [ 𝑧 / 𝑦 ] ∀ 𝑥 𝜑 ↔ ∀ 𝑦 ∀ 𝑥 ( 𝑦 = 𝑧 → 𝜑 ) ) ) |
11 |
|
nfnae |
⊢ Ⅎ 𝑥 ¬ ∀ 𝑦 𝑦 = 𝑧 |
12 |
|
sb4b |
⊢ ( ¬ ∀ 𝑦 𝑦 = 𝑧 → ( [ 𝑧 / 𝑦 ] 𝜑 ↔ ∀ 𝑦 ( 𝑦 = 𝑧 → 𝜑 ) ) ) |
13 |
11 12
|
albid |
⊢ ( ¬ ∀ 𝑦 𝑦 = 𝑧 → ( ∀ 𝑥 [ 𝑧 / 𝑦 ] 𝜑 ↔ ∀ 𝑥 ∀ 𝑦 ( 𝑦 = 𝑧 → 𝜑 ) ) ) |
14 |
|
alcom |
⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝑦 = 𝑧 → 𝜑 ) ↔ ∀ 𝑦 ∀ 𝑥 ( 𝑦 = 𝑧 → 𝜑 ) ) |
15 |
13 14
|
bitrdi |
⊢ ( ¬ ∀ 𝑦 𝑦 = 𝑧 → ( ∀ 𝑥 [ 𝑧 / 𝑦 ] 𝜑 ↔ ∀ 𝑦 ∀ 𝑥 ( 𝑦 = 𝑧 → 𝜑 ) ) ) |
16 |
15
|
adantl |
⊢ ( ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) ∧ ¬ ∀ 𝑦 𝑦 = 𝑧 ) → ( ∀ 𝑥 [ 𝑧 / 𝑦 ] 𝜑 ↔ ∀ 𝑦 ∀ 𝑥 ( 𝑦 = 𝑧 → 𝜑 ) ) ) |
17 |
10 16
|
bitr4d |
⊢ ( ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) ∧ ¬ ∀ 𝑦 𝑦 = 𝑧 ) → ( [ 𝑧 / 𝑦 ] ∀ 𝑥 𝜑 ↔ ∀ 𝑥 [ 𝑧 / 𝑦 ] 𝜑 ) ) |
18 |
17
|
ex |
⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) → ( ¬ ∀ 𝑦 𝑦 = 𝑧 → ( [ 𝑧 / 𝑦 ] ∀ 𝑥 𝜑 ↔ ∀ 𝑥 [ 𝑧 / 𝑦 ] 𝜑 ) ) ) |
19 |
|
sbequ12 |
⊢ ( 𝑦 = 𝑧 → ( ∀ 𝑥 𝜑 ↔ [ 𝑧 / 𝑦 ] ∀ 𝑥 𝜑 ) ) |
20 |
19
|
sps |
⊢ ( ∀ 𝑦 𝑦 = 𝑧 → ( ∀ 𝑥 𝜑 ↔ [ 𝑧 / 𝑦 ] ∀ 𝑥 𝜑 ) ) |
21 |
|
sbequ12 |
⊢ ( 𝑦 = 𝑧 → ( 𝜑 ↔ [ 𝑧 / 𝑦 ] 𝜑 ) ) |
22 |
21
|
sps |
⊢ ( ∀ 𝑦 𝑦 = 𝑧 → ( 𝜑 ↔ [ 𝑧 / 𝑦 ] 𝜑 ) ) |
23 |
22
|
dral2 |
⊢ ( ∀ 𝑦 𝑦 = 𝑧 → ( ∀ 𝑥 𝜑 ↔ ∀ 𝑥 [ 𝑧 / 𝑦 ] 𝜑 ) ) |
24 |
20 23
|
bitr3d |
⊢ ( ∀ 𝑦 𝑦 = 𝑧 → ( [ 𝑧 / 𝑦 ] ∀ 𝑥 𝜑 ↔ ∀ 𝑥 [ 𝑧 / 𝑦 ] 𝜑 ) ) |
25 |
18 24
|
pm2.61d2 |
⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) → ( [ 𝑧 / 𝑦 ] ∀ 𝑥 𝜑 ↔ ∀ 𝑥 [ 𝑧 / 𝑦 ] 𝜑 ) ) |