Step |
Hyp |
Ref |
Expression |
1 |
|
zfpow |
⊢ ∃ 𝑤 ∀ 𝑦 ( ∀ 𝑤 ( 𝑤 ∈ 𝑦 → 𝑤 ∈ 𝑧 ) → 𝑦 ∈ 𝑤 ) |
2 |
|
19.8a |
⊢ ( 𝑤 ∈ 𝑦 → ∃ 𝑧 𝑤 ∈ 𝑦 ) |
3 |
|
sp |
⊢ ( ∀ 𝑦 𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑧 ) |
4 |
2 3
|
imim12i |
⊢ ( ( ∃ 𝑧 𝑤 ∈ 𝑦 → ∀ 𝑦 𝑤 ∈ 𝑧 ) → ( 𝑤 ∈ 𝑦 → 𝑤 ∈ 𝑧 ) ) |
5 |
4
|
alimi |
⊢ ( ∀ 𝑤 ( ∃ 𝑧 𝑤 ∈ 𝑦 → ∀ 𝑦 𝑤 ∈ 𝑧 ) → ∀ 𝑤 ( 𝑤 ∈ 𝑦 → 𝑤 ∈ 𝑧 ) ) |
6 |
5
|
imim1i |
⊢ ( ( ∀ 𝑤 ( 𝑤 ∈ 𝑦 → 𝑤 ∈ 𝑧 ) → 𝑦 ∈ 𝑤 ) → ( ∀ 𝑤 ( ∃ 𝑧 𝑤 ∈ 𝑦 → ∀ 𝑦 𝑤 ∈ 𝑧 ) → 𝑦 ∈ 𝑤 ) ) |
7 |
6
|
alimi |
⊢ ( ∀ 𝑦 ( ∀ 𝑤 ( 𝑤 ∈ 𝑦 → 𝑤 ∈ 𝑧 ) → 𝑦 ∈ 𝑤 ) → ∀ 𝑦 ( ∀ 𝑤 ( ∃ 𝑧 𝑤 ∈ 𝑦 → ∀ 𝑦 𝑤 ∈ 𝑧 ) → 𝑦 ∈ 𝑤 ) ) |
8 |
1 7
|
eximii |
⊢ ∃ 𝑤 ∀ 𝑦 ( ∀ 𝑤 ( ∃ 𝑧 𝑤 ∈ 𝑦 → ∀ 𝑦 𝑤 ∈ 𝑧 ) → 𝑦 ∈ 𝑤 ) |
9 |
|
nfnae |
⊢ Ⅎ 𝑥 ¬ ∀ 𝑥 𝑥 = 𝑦 |
10 |
|
nfnae |
⊢ Ⅎ 𝑥 ¬ ∀ 𝑥 𝑥 = 𝑧 |
11 |
9 10
|
nfan |
⊢ Ⅎ 𝑥 ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) |
12 |
|
nfnae |
⊢ Ⅎ 𝑦 ¬ ∀ 𝑥 𝑥 = 𝑦 |
13 |
|
nfnae |
⊢ Ⅎ 𝑦 ¬ ∀ 𝑥 𝑥 = 𝑧 |
14 |
12 13
|
nfan |
⊢ Ⅎ 𝑦 ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) |
15 |
|
nfv |
⊢ Ⅎ 𝑤 ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) |
16 |
|
nfnae |
⊢ Ⅎ 𝑧 ¬ ∀ 𝑥 𝑥 = 𝑦 |
17 |
|
nfcvd |
⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑥 𝑤 ) |
18 |
|
nfcvf |
⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑥 𝑦 ) |
19 |
17 18
|
nfeld |
⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑥 𝑤 ∈ 𝑦 ) |
20 |
16 19
|
nfexd |
⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑥 ∃ 𝑧 𝑤 ∈ 𝑦 ) |
21 |
20
|
adantr |
⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) → Ⅎ 𝑥 ∃ 𝑧 𝑤 ∈ 𝑦 ) |
22 |
|
nfcvd |
⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑧 → Ⅎ 𝑥 𝑤 ) |
23 |
|
nfcvf |
⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑧 → Ⅎ 𝑥 𝑧 ) |
24 |
22 23
|
nfeld |
⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑧 → Ⅎ 𝑥 𝑤 ∈ 𝑧 ) |
25 |
13 24
|
nfald |
⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑧 → Ⅎ 𝑥 ∀ 𝑦 𝑤 ∈ 𝑧 ) |
26 |
25
|
adantl |
⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) → Ⅎ 𝑥 ∀ 𝑦 𝑤 ∈ 𝑧 ) |
27 |
21 26
|
nfimd |
⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) → Ⅎ 𝑥 ( ∃ 𝑧 𝑤 ∈ 𝑦 → ∀ 𝑦 𝑤 ∈ 𝑧 ) ) |
28 |
15 27
|
nfald |
⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) → Ⅎ 𝑥 ∀ 𝑤 ( ∃ 𝑧 𝑤 ∈ 𝑦 → ∀ 𝑦 𝑤 ∈ 𝑧 ) ) |
29 |
18 17
|
nfeld |
⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑥 𝑦 ∈ 𝑤 ) |
30 |
29
|
adantr |
⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) → Ⅎ 𝑥 𝑦 ∈ 𝑤 ) |
31 |
28 30
|
nfimd |
⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) → Ⅎ 𝑥 ( ∀ 𝑤 ( ∃ 𝑧 𝑤 ∈ 𝑦 → ∀ 𝑦 𝑤 ∈ 𝑧 ) → 𝑦 ∈ 𝑤 ) ) |
32 |
14 31
|
nfald |
⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) → Ⅎ 𝑥 ∀ 𝑦 ( ∀ 𝑤 ( ∃ 𝑧 𝑤 ∈ 𝑦 → ∀ 𝑦 𝑤 ∈ 𝑧 ) → 𝑦 ∈ 𝑤 ) ) |
33 |
|
nfeqf2 |
⊢ ( ¬ ∀ 𝑦 𝑦 = 𝑥 → Ⅎ 𝑦 𝑤 = 𝑥 ) |
34 |
33
|
naecoms |
⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑦 𝑤 = 𝑥 ) |
35 |
34
|
adantr |
⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) → Ⅎ 𝑦 𝑤 = 𝑥 ) |
36 |
14 35
|
nfan1 |
⊢ Ⅎ 𝑦 ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) ∧ 𝑤 = 𝑥 ) |
37 |
|
nfnae |
⊢ Ⅎ 𝑧 ¬ ∀ 𝑥 𝑥 = 𝑧 |
38 |
|
nfeqf2 |
⊢ ( ¬ ∀ 𝑧 𝑧 = 𝑥 → Ⅎ 𝑧 𝑤 = 𝑥 ) |
39 |
38
|
naecoms |
⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑧 → Ⅎ 𝑧 𝑤 = 𝑥 ) |
40 |
37 39
|
nfan1 |
⊢ Ⅎ 𝑧 ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ 𝑤 = 𝑥 ) |
41 |
|
elequ1 |
⊢ ( 𝑤 = 𝑥 → ( 𝑤 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦 ) ) |
42 |
41
|
adantl |
⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ 𝑤 = 𝑥 ) → ( 𝑤 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦 ) ) |
43 |
40 42
|
exbid |
⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ 𝑤 = 𝑥 ) → ( ∃ 𝑧 𝑤 ∈ 𝑦 ↔ ∃ 𝑧 𝑥 ∈ 𝑦 ) ) |
44 |
43
|
adantll |
⊢ ( ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) ∧ 𝑤 = 𝑥 ) → ( ∃ 𝑧 𝑤 ∈ 𝑦 ↔ ∃ 𝑧 𝑥 ∈ 𝑦 ) ) |
45 |
12 34
|
nfan1 |
⊢ Ⅎ 𝑦 ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ 𝑤 = 𝑥 ) |
46 |
|
elequ1 |
⊢ ( 𝑤 = 𝑥 → ( 𝑤 ∈ 𝑧 ↔ 𝑥 ∈ 𝑧 ) ) |
47 |
46
|
adantl |
⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ 𝑤 = 𝑥 ) → ( 𝑤 ∈ 𝑧 ↔ 𝑥 ∈ 𝑧 ) ) |
48 |
45 47
|
albid |
⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ 𝑤 = 𝑥 ) → ( ∀ 𝑦 𝑤 ∈ 𝑧 ↔ ∀ 𝑦 𝑥 ∈ 𝑧 ) ) |
49 |
48
|
adantlr |
⊢ ( ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) ∧ 𝑤 = 𝑥 ) → ( ∀ 𝑦 𝑤 ∈ 𝑧 ↔ ∀ 𝑦 𝑥 ∈ 𝑧 ) ) |
50 |
44 49
|
imbi12d |
⊢ ( ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) ∧ 𝑤 = 𝑥 ) → ( ( ∃ 𝑧 𝑤 ∈ 𝑦 → ∀ 𝑦 𝑤 ∈ 𝑧 ) ↔ ( ∃ 𝑧 𝑥 ∈ 𝑦 → ∀ 𝑦 𝑥 ∈ 𝑧 ) ) ) |
51 |
50
|
ex |
⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) → ( 𝑤 = 𝑥 → ( ( ∃ 𝑧 𝑤 ∈ 𝑦 → ∀ 𝑦 𝑤 ∈ 𝑧 ) ↔ ( ∃ 𝑧 𝑥 ∈ 𝑦 → ∀ 𝑦 𝑥 ∈ 𝑧 ) ) ) ) |
52 |
11 27 51
|
cbvald |
⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) → ( ∀ 𝑤 ( ∃ 𝑧 𝑤 ∈ 𝑦 → ∀ 𝑦 𝑤 ∈ 𝑧 ) ↔ ∀ 𝑥 ( ∃ 𝑧 𝑥 ∈ 𝑦 → ∀ 𝑦 𝑥 ∈ 𝑧 ) ) ) |
53 |
52
|
adantr |
⊢ ( ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) ∧ 𝑤 = 𝑥 ) → ( ∀ 𝑤 ( ∃ 𝑧 𝑤 ∈ 𝑦 → ∀ 𝑦 𝑤 ∈ 𝑧 ) ↔ ∀ 𝑥 ( ∃ 𝑧 𝑥 ∈ 𝑦 → ∀ 𝑦 𝑥 ∈ 𝑧 ) ) ) |
54 |
|
elequ2 |
⊢ ( 𝑤 = 𝑥 → ( 𝑦 ∈ 𝑤 ↔ 𝑦 ∈ 𝑥 ) ) |
55 |
54
|
adantl |
⊢ ( ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) ∧ 𝑤 = 𝑥 ) → ( 𝑦 ∈ 𝑤 ↔ 𝑦 ∈ 𝑥 ) ) |
56 |
53 55
|
imbi12d |
⊢ ( ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) ∧ 𝑤 = 𝑥 ) → ( ( ∀ 𝑤 ( ∃ 𝑧 𝑤 ∈ 𝑦 → ∀ 𝑦 𝑤 ∈ 𝑧 ) → 𝑦 ∈ 𝑤 ) ↔ ( ∀ 𝑥 ( ∃ 𝑧 𝑥 ∈ 𝑦 → ∀ 𝑦 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) ) ) |
57 |
36 56
|
albid |
⊢ ( ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) ∧ 𝑤 = 𝑥 ) → ( ∀ 𝑦 ( ∀ 𝑤 ( ∃ 𝑧 𝑤 ∈ 𝑦 → ∀ 𝑦 𝑤 ∈ 𝑧 ) → 𝑦 ∈ 𝑤 ) ↔ ∀ 𝑦 ( ∀ 𝑥 ( ∃ 𝑧 𝑥 ∈ 𝑦 → ∀ 𝑦 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) ) ) |
58 |
57
|
ex |
⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) → ( 𝑤 = 𝑥 → ( ∀ 𝑦 ( ∀ 𝑤 ( ∃ 𝑧 𝑤 ∈ 𝑦 → ∀ 𝑦 𝑤 ∈ 𝑧 ) → 𝑦 ∈ 𝑤 ) ↔ ∀ 𝑦 ( ∀ 𝑥 ( ∃ 𝑧 𝑥 ∈ 𝑦 → ∀ 𝑦 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) ) ) ) |
59 |
11 32 58
|
cbvexd |
⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) → ( ∃ 𝑤 ∀ 𝑦 ( ∀ 𝑤 ( ∃ 𝑧 𝑤 ∈ 𝑦 → ∀ 𝑦 𝑤 ∈ 𝑧 ) → 𝑦 ∈ 𝑤 ) ↔ ∃ 𝑥 ∀ 𝑦 ( ∀ 𝑥 ( ∃ 𝑧 𝑥 ∈ 𝑦 → ∀ 𝑦 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) ) ) |
60 |
8 59
|
mpbii |
⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) → ∃ 𝑥 ∀ 𝑦 ( ∀ 𝑥 ( ∃ 𝑧 𝑥 ∈ 𝑦 → ∀ 𝑦 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) ) |
61 |
60
|
ex |
⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( ¬ ∀ 𝑥 𝑥 = 𝑧 → ∃ 𝑥 ∀ 𝑦 ( ∀ 𝑥 ( ∃ 𝑧 𝑥 ∈ 𝑦 → ∀ 𝑦 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) ) ) |