Step |
Hyp |
Ref |
Expression |
1 |
|
nfe1 |
⊢ Ⅎ 𝑦 ∃ 𝑦 ∀ 𝑧 ( ∀ 𝑦 𝜑 → 𝑧 = 𝑦 ) |
2 |
|
nfv |
⊢ Ⅎ 𝑦 𝑧 ∈ 𝑤 |
3 |
|
nfv |
⊢ Ⅎ 𝑦 𝑤 ∈ 𝑥 |
4 |
|
nfa1 |
⊢ Ⅎ 𝑦 ∀ 𝑦 ∀ 𝑦 𝜑 |
5 |
3 4
|
nfan |
⊢ Ⅎ 𝑦 ( 𝑤 ∈ 𝑥 ∧ ∀ 𝑦 ∀ 𝑦 𝜑 ) |
6 |
5
|
nfex |
⊢ Ⅎ 𝑦 ∃ 𝑤 ( 𝑤 ∈ 𝑥 ∧ ∀ 𝑦 ∀ 𝑦 𝜑 ) |
7 |
2 6
|
nfbi |
⊢ Ⅎ 𝑦 ( 𝑧 ∈ 𝑤 ↔ ∃ 𝑤 ( 𝑤 ∈ 𝑥 ∧ ∀ 𝑦 ∀ 𝑦 𝜑 ) ) |
8 |
7
|
nfal |
⊢ Ⅎ 𝑦 ∀ 𝑧 ( 𝑧 ∈ 𝑤 ↔ ∃ 𝑤 ( 𝑤 ∈ 𝑥 ∧ ∀ 𝑦 ∀ 𝑦 𝜑 ) ) |
9 |
1 8
|
nfim |
⊢ Ⅎ 𝑦 ( ∃ 𝑦 ∀ 𝑧 ( ∀ 𝑦 𝜑 → 𝑧 = 𝑦 ) → ∀ 𝑧 ( 𝑧 ∈ 𝑤 ↔ ∃ 𝑤 ( 𝑤 ∈ 𝑥 ∧ ∀ 𝑦 ∀ 𝑦 𝜑 ) ) ) |
10 |
9
|
nfex |
⊢ Ⅎ 𝑦 ∃ 𝑤 ( ∃ 𝑦 ∀ 𝑧 ( ∀ 𝑦 𝜑 → 𝑧 = 𝑦 ) → ∀ 𝑧 ( 𝑧 ∈ 𝑤 ↔ ∃ 𝑤 ( 𝑤 ∈ 𝑥 ∧ ∀ 𝑦 ∀ 𝑦 𝜑 ) ) ) |
11 |
|
elequ2 |
⊢ ( 𝑦 = 𝑥 → ( 𝑤 ∈ 𝑦 ↔ 𝑤 ∈ 𝑥 ) ) |
12 |
11
|
anbi1d |
⊢ ( 𝑦 = 𝑥 → ( ( 𝑤 ∈ 𝑦 ∧ ∀ 𝑦 ∀ 𝑦 𝜑 ) ↔ ( 𝑤 ∈ 𝑥 ∧ ∀ 𝑦 ∀ 𝑦 𝜑 ) ) ) |
13 |
12
|
exbidv |
⊢ ( 𝑦 = 𝑥 → ( ∃ 𝑤 ( 𝑤 ∈ 𝑦 ∧ ∀ 𝑦 ∀ 𝑦 𝜑 ) ↔ ∃ 𝑤 ( 𝑤 ∈ 𝑥 ∧ ∀ 𝑦 ∀ 𝑦 𝜑 ) ) ) |
14 |
13
|
bibi2d |
⊢ ( 𝑦 = 𝑥 → ( ( 𝑧 ∈ 𝑤 ↔ ∃ 𝑤 ( 𝑤 ∈ 𝑦 ∧ ∀ 𝑦 ∀ 𝑦 𝜑 ) ) ↔ ( 𝑧 ∈ 𝑤 ↔ ∃ 𝑤 ( 𝑤 ∈ 𝑥 ∧ ∀ 𝑦 ∀ 𝑦 𝜑 ) ) ) ) |
15 |
14
|
albidv |
⊢ ( 𝑦 = 𝑥 → ( ∀ 𝑧 ( 𝑧 ∈ 𝑤 ↔ ∃ 𝑤 ( 𝑤 ∈ 𝑦 ∧ ∀ 𝑦 ∀ 𝑦 𝜑 ) ) ↔ ∀ 𝑧 ( 𝑧 ∈ 𝑤 ↔ ∃ 𝑤 ( 𝑤 ∈ 𝑥 ∧ ∀ 𝑦 ∀ 𝑦 𝜑 ) ) ) ) |
16 |
15
|
imbi2d |
⊢ ( 𝑦 = 𝑥 → ( ( ∃ 𝑦 ∀ 𝑧 ( ∀ 𝑦 𝜑 → 𝑧 = 𝑦 ) → ∀ 𝑧 ( 𝑧 ∈ 𝑤 ↔ ∃ 𝑤 ( 𝑤 ∈ 𝑦 ∧ ∀ 𝑦 ∀ 𝑦 𝜑 ) ) ) ↔ ( ∃ 𝑦 ∀ 𝑧 ( ∀ 𝑦 𝜑 → 𝑧 = 𝑦 ) → ∀ 𝑧 ( 𝑧 ∈ 𝑤 ↔ ∃ 𝑤 ( 𝑤 ∈ 𝑥 ∧ ∀ 𝑦 ∀ 𝑦 𝜑 ) ) ) ) ) |
17 |
16
|
exbidv |
⊢ ( 𝑦 = 𝑥 → ( ∃ 𝑤 ( ∃ 𝑦 ∀ 𝑧 ( ∀ 𝑦 𝜑 → 𝑧 = 𝑦 ) → ∀ 𝑧 ( 𝑧 ∈ 𝑤 ↔ ∃ 𝑤 ( 𝑤 ∈ 𝑦 ∧ ∀ 𝑦 ∀ 𝑦 𝜑 ) ) ) ↔ ∃ 𝑤 ( ∃ 𝑦 ∀ 𝑧 ( ∀ 𝑦 𝜑 → 𝑧 = 𝑦 ) → ∀ 𝑧 ( 𝑧 ∈ 𝑤 ↔ ∃ 𝑤 ( 𝑤 ∈ 𝑥 ∧ ∀ 𝑦 ∀ 𝑦 𝜑 ) ) ) ) ) |
18 |
|
axrepnd |
⊢ ∃ 𝑤 ( ∃ 𝑦 ∀ 𝑧 ( ∀ 𝑦 𝜑 → 𝑧 = 𝑦 ) → ∀ 𝑧 ( ∀ 𝑦 𝑧 ∈ 𝑤 ↔ ∃ 𝑤 ( ∀ 𝑧 𝑤 ∈ 𝑦 ∧ ∀ 𝑦 ∀ 𝑦 𝜑 ) ) ) |
19 |
|
19.3v |
⊢ ( ∀ 𝑦 𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑤 ) |
20 |
|
19.3v |
⊢ ( ∀ 𝑧 𝑤 ∈ 𝑦 ↔ 𝑤 ∈ 𝑦 ) |
21 |
20
|
anbi1i |
⊢ ( ( ∀ 𝑧 𝑤 ∈ 𝑦 ∧ ∀ 𝑦 ∀ 𝑦 𝜑 ) ↔ ( 𝑤 ∈ 𝑦 ∧ ∀ 𝑦 ∀ 𝑦 𝜑 ) ) |
22 |
21
|
exbii |
⊢ ( ∃ 𝑤 ( ∀ 𝑧 𝑤 ∈ 𝑦 ∧ ∀ 𝑦 ∀ 𝑦 𝜑 ) ↔ ∃ 𝑤 ( 𝑤 ∈ 𝑦 ∧ ∀ 𝑦 ∀ 𝑦 𝜑 ) ) |
23 |
19 22
|
bibi12i |
⊢ ( ( ∀ 𝑦 𝑧 ∈ 𝑤 ↔ ∃ 𝑤 ( ∀ 𝑧 𝑤 ∈ 𝑦 ∧ ∀ 𝑦 ∀ 𝑦 𝜑 ) ) ↔ ( 𝑧 ∈ 𝑤 ↔ ∃ 𝑤 ( 𝑤 ∈ 𝑦 ∧ ∀ 𝑦 ∀ 𝑦 𝜑 ) ) ) |
24 |
23
|
albii |
⊢ ( ∀ 𝑧 ( ∀ 𝑦 𝑧 ∈ 𝑤 ↔ ∃ 𝑤 ( ∀ 𝑧 𝑤 ∈ 𝑦 ∧ ∀ 𝑦 ∀ 𝑦 𝜑 ) ) ↔ ∀ 𝑧 ( 𝑧 ∈ 𝑤 ↔ ∃ 𝑤 ( 𝑤 ∈ 𝑦 ∧ ∀ 𝑦 ∀ 𝑦 𝜑 ) ) ) |
25 |
24
|
imbi2i |
⊢ ( ( ∃ 𝑦 ∀ 𝑧 ( ∀ 𝑦 𝜑 → 𝑧 = 𝑦 ) → ∀ 𝑧 ( ∀ 𝑦 𝑧 ∈ 𝑤 ↔ ∃ 𝑤 ( ∀ 𝑧 𝑤 ∈ 𝑦 ∧ ∀ 𝑦 ∀ 𝑦 𝜑 ) ) ) ↔ ( ∃ 𝑦 ∀ 𝑧 ( ∀ 𝑦 𝜑 → 𝑧 = 𝑦 ) → ∀ 𝑧 ( 𝑧 ∈ 𝑤 ↔ ∃ 𝑤 ( 𝑤 ∈ 𝑦 ∧ ∀ 𝑦 ∀ 𝑦 𝜑 ) ) ) ) |
26 |
25
|
exbii |
⊢ ( ∃ 𝑤 ( ∃ 𝑦 ∀ 𝑧 ( ∀ 𝑦 𝜑 → 𝑧 = 𝑦 ) → ∀ 𝑧 ( ∀ 𝑦 𝑧 ∈ 𝑤 ↔ ∃ 𝑤 ( ∀ 𝑧 𝑤 ∈ 𝑦 ∧ ∀ 𝑦 ∀ 𝑦 𝜑 ) ) ) ↔ ∃ 𝑤 ( ∃ 𝑦 ∀ 𝑧 ( ∀ 𝑦 𝜑 → 𝑧 = 𝑦 ) → ∀ 𝑧 ( 𝑧 ∈ 𝑤 ↔ ∃ 𝑤 ( 𝑤 ∈ 𝑦 ∧ ∀ 𝑦 ∀ 𝑦 𝜑 ) ) ) ) |
27 |
18 26
|
mpbi |
⊢ ∃ 𝑤 ( ∃ 𝑦 ∀ 𝑧 ( ∀ 𝑦 𝜑 → 𝑧 = 𝑦 ) → ∀ 𝑧 ( 𝑧 ∈ 𝑤 ↔ ∃ 𝑤 ( 𝑤 ∈ 𝑦 ∧ ∀ 𝑦 ∀ 𝑦 𝜑 ) ) ) |
28 |
10 17 27
|
chvar |
⊢ ∃ 𝑤 ( ∃ 𝑦 ∀ 𝑧 ( ∀ 𝑦 𝜑 → 𝑧 = 𝑦 ) → ∀ 𝑧 ( 𝑧 ∈ 𝑤 ↔ ∃ 𝑤 ( 𝑤 ∈ 𝑥 ∧ ∀ 𝑦 ∀ 𝑦 𝜑 ) ) ) |
29 |
28
|
19.35i |
⊢ ( ∀ 𝑤 ∃ 𝑦 ∀ 𝑧 ( ∀ 𝑦 𝜑 → 𝑧 = 𝑦 ) → ∃ 𝑤 ∀ 𝑧 ( 𝑧 ∈ 𝑤 ↔ ∃ 𝑤 ( 𝑤 ∈ 𝑥 ∧ ∀ 𝑦 ∀ 𝑦 𝜑 ) ) ) |
30 |
|
nfv |
⊢ Ⅎ 𝑤 𝑧 ∈ 𝑦 |
31 |
|
nfe1 |
⊢ Ⅎ 𝑤 ∃ 𝑤 ( 𝑤 ∈ 𝑥 ∧ ∀ 𝑦 𝜑 ) |
32 |
30 31
|
nfbi |
⊢ Ⅎ 𝑤 ( 𝑧 ∈ 𝑦 ↔ ∃ 𝑤 ( 𝑤 ∈ 𝑥 ∧ ∀ 𝑦 𝜑 ) ) |
33 |
32
|
nfal |
⊢ Ⅎ 𝑤 ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ ∃ 𝑤 ( 𝑤 ∈ 𝑥 ∧ ∀ 𝑦 𝜑 ) ) |
34 |
|
elequ2 |
⊢ ( 𝑤 = 𝑦 → ( 𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦 ) ) |
35 |
|
nfa1 |
⊢ Ⅎ 𝑦 ∀ 𝑦 𝜑 |
36 |
35
|
19.3 |
⊢ ( ∀ 𝑦 ∀ 𝑦 𝜑 ↔ ∀ 𝑦 𝜑 ) |
37 |
36
|
anbi2i |
⊢ ( ( 𝑤 ∈ 𝑥 ∧ ∀ 𝑦 ∀ 𝑦 𝜑 ) ↔ ( 𝑤 ∈ 𝑥 ∧ ∀ 𝑦 𝜑 ) ) |
38 |
37
|
exbii |
⊢ ( ∃ 𝑤 ( 𝑤 ∈ 𝑥 ∧ ∀ 𝑦 ∀ 𝑦 𝜑 ) ↔ ∃ 𝑤 ( 𝑤 ∈ 𝑥 ∧ ∀ 𝑦 𝜑 ) ) |
39 |
38
|
a1i |
⊢ ( 𝑤 = 𝑦 → ( ∃ 𝑤 ( 𝑤 ∈ 𝑥 ∧ ∀ 𝑦 ∀ 𝑦 𝜑 ) ↔ ∃ 𝑤 ( 𝑤 ∈ 𝑥 ∧ ∀ 𝑦 𝜑 ) ) ) |
40 |
34 39
|
bibi12d |
⊢ ( 𝑤 = 𝑦 → ( ( 𝑧 ∈ 𝑤 ↔ ∃ 𝑤 ( 𝑤 ∈ 𝑥 ∧ ∀ 𝑦 ∀ 𝑦 𝜑 ) ) ↔ ( 𝑧 ∈ 𝑦 ↔ ∃ 𝑤 ( 𝑤 ∈ 𝑥 ∧ ∀ 𝑦 𝜑 ) ) ) ) |
41 |
40
|
albidv |
⊢ ( 𝑤 = 𝑦 → ( ∀ 𝑧 ( 𝑧 ∈ 𝑤 ↔ ∃ 𝑤 ( 𝑤 ∈ 𝑥 ∧ ∀ 𝑦 ∀ 𝑦 𝜑 ) ) ↔ ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ ∃ 𝑤 ( 𝑤 ∈ 𝑥 ∧ ∀ 𝑦 𝜑 ) ) ) ) |
42 |
8 33 41
|
cbvexv1 |
⊢ ( ∃ 𝑤 ∀ 𝑧 ( 𝑧 ∈ 𝑤 ↔ ∃ 𝑤 ( 𝑤 ∈ 𝑥 ∧ ∀ 𝑦 ∀ 𝑦 𝜑 ) ) ↔ ∃ 𝑦 ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ ∃ 𝑤 ( 𝑤 ∈ 𝑥 ∧ ∀ 𝑦 𝜑 ) ) ) |
43 |
29 42
|
sylib |
⊢ ( ∀ 𝑤 ∃ 𝑦 ∀ 𝑧 ( ∀ 𝑦 𝜑 → 𝑧 = 𝑦 ) → ∃ 𝑦 ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ ∃ 𝑤 ( 𝑤 ∈ 𝑥 ∧ ∀ 𝑦 𝜑 ) ) ) |