Step |
Hyp |
Ref |
Expression |
1 |
|
rmo3f.1 |
⊢ Ⅎ 𝑥 𝐴 |
2 |
|
rmo3f.2 |
⊢ Ⅎ 𝑦 𝐴 |
3 |
|
rmo3f.3 |
⊢ Ⅎ 𝑦 𝜑 |
4 |
|
df-rmo |
⊢ ( ∃* 𝑥 ∈ 𝐴 𝜑 ↔ ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) |
5 |
|
sban |
⊢ ( [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ( [ 𝑦 / 𝑥 ] 𝑥 ∈ 𝐴 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) ) |
6 |
1
|
clelsb1fw |
⊢ ( [ 𝑦 / 𝑥 ] 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴 ) |
7 |
5 6
|
bianbi |
⊢ ( [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ( 𝑦 ∈ 𝐴 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) ) |
8 |
7
|
anbi2i |
⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ∧ [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) ↔ ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ∧ ( 𝑦 ∈ 𝐴 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) ) ) |
9 |
|
an4 |
⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ∧ ( 𝑦 ∈ 𝐴 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) ) ↔ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ∧ ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) ) ) |
10 |
|
ancom |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ↔ ( 𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴 ) ) |
11 |
10
|
anbi1i |
⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ∧ ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) ) ↔ ( ( 𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴 ) ∧ ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) ) ) |
12 |
8 9 11
|
3bitri |
⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ∧ [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) ↔ ( ( 𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴 ) ∧ ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) ) ) |
13 |
12
|
imbi1i |
⊢ ( ( ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ∧ [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) → 𝑥 = 𝑦 ) ↔ ( ( ( 𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴 ) ∧ ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) ) → 𝑥 = 𝑦 ) ) |
14 |
|
impexp |
⊢ ( ( ( ( 𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴 ) ∧ ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) ) → 𝑥 = 𝑦 ) ↔ ( ( 𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) → 𝑥 = 𝑦 ) ) ) |
15 |
|
impexp |
⊢ ( ( ( 𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) → 𝑥 = 𝑦 ) ) ↔ ( 𝑦 ∈ 𝐴 → ( 𝑥 ∈ 𝐴 → ( ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) → 𝑥 = 𝑦 ) ) ) ) |
16 |
13 14 15
|
3bitri |
⊢ ( ( ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ∧ [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) → 𝑥 = 𝑦 ) ↔ ( 𝑦 ∈ 𝐴 → ( 𝑥 ∈ 𝐴 → ( ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) → 𝑥 = 𝑦 ) ) ) ) |
17 |
16
|
albii |
⊢ ( ∀ 𝑦 ( ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ∧ [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) → 𝑥 = 𝑦 ) ↔ ∀ 𝑦 ( 𝑦 ∈ 𝐴 → ( 𝑥 ∈ 𝐴 → ( ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) → 𝑥 = 𝑦 ) ) ) ) |
18 |
|
df-ral |
⊢ ( ∀ 𝑦 ∈ 𝐴 ( 𝑥 ∈ 𝐴 → ( ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) → 𝑥 = 𝑦 ) ) ↔ ∀ 𝑦 ( 𝑦 ∈ 𝐴 → ( 𝑥 ∈ 𝐴 → ( ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) → 𝑥 = 𝑦 ) ) ) ) |
19 |
2
|
nfcri |
⊢ Ⅎ 𝑦 𝑥 ∈ 𝐴 |
20 |
19
|
r19.21 |
⊢ ( ∀ 𝑦 ∈ 𝐴 ( 𝑥 ∈ 𝐴 → ( ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) → 𝑥 = 𝑦 ) ) ↔ ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝐴 ( ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) → 𝑥 = 𝑦 ) ) ) |
21 |
17 18 20
|
3bitr2i |
⊢ ( ∀ 𝑦 ( ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ∧ [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) → 𝑥 = 𝑦 ) ↔ ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝐴 ( ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) → 𝑥 = 𝑦 ) ) ) |
22 |
21
|
albii |
⊢ ( ∀ 𝑥 ∀ 𝑦 ( ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ∧ [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) → 𝑥 = 𝑦 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝐴 ( ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) → 𝑥 = 𝑦 ) ) ) |
23 |
19 3
|
nfan |
⊢ Ⅎ 𝑦 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) |
24 |
23
|
mo3 |
⊢ ( ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ∀ 𝑥 ∀ 𝑦 ( ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ∧ [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) → 𝑥 = 𝑦 ) ) |
25 |
|
df-ral |
⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) → 𝑥 = 𝑦 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝐴 ( ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) → 𝑥 = 𝑦 ) ) ) |
26 |
22 24 25
|
3bitr4i |
⊢ ( ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) → 𝑥 = 𝑦 ) ) |
27 |
4 26
|
bitri |
⊢ ( ∃* 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) → 𝑥 = 𝑦 ) ) |