Step |
Hyp |
Ref |
Expression |
1 |
|
axrepndlem2 |
⊢ ( ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) ∧ ¬ ∀ 𝑦 𝑦 = 𝑧 ) → ∃ 𝑥 ( ∃ 𝑦 ∀ 𝑧 ( 𝜑 → 𝑧 = 𝑦 ) → ∀ 𝑧 ( 𝑧 ∈ 𝑥 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝑦 ∧ ∀ 𝑦 𝜑 ) ) ) ) |
2 |
|
nfnae |
⊢ Ⅎ 𝑥 ¬ ∀ 𝑥 𝑥 = 𝑦 |
3 |
|
nfnae |
⊢ Ⅎ 𝑥 ¬ ∀ 𝑥 𝑥 = 𝑧 |
4 |
2 3
|
nfan |
⊢ Ⅎ 𝑥 ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) |
5 |
|
nfnae |
⊢ Ⅎ 𝑥 ¬ ∀ 𝑦 𝑦 = 𝑧 |
6 |
4 5
|
nfan |
⊢ Ⅎ 𝑥 ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) ∧ ¬ ∀ 𝑦 𝑦 = 𝑧 ) |
7 |
|
nfnae |
⊢ Ⅎ 𝑧 ¬ ∀ 𝑥 𝑥 = 𝑦 |
8 |
|
nfnae |
⊢ Ⅎ 𝑧 ¬ ∀ 𝑥 𝑥 = 𝑧 |
9 |
7 8
|
nfan |
⊢ Ⅎ 𝑧 ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) |
10 |
|
nfnae |
⊢ Ⅎ 𝑧 ¬ ∀ 𝑦 𝑦 = 𝑧 |
11 |
9 10
|
nfan |
⊢ Ⅎ 𝑧 ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) ∧ ¬ ∀ 𝑦 𝑦 = 𝑧 ) |
12 |
|
nfcvf |
⊢ ( ¬ ∀ 𝑦 𝑦 = 𝑧 → Ⅎ 𝑦 𝑧 ) |
13 |
12
|
adantl |
⊢ ( ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) ∧ ¬ ∀ 𝑦 𝑦 = 𝑧 ) → Ⅎ 𝑦 𝑧 ) |
14 |
|
nfcvf2 |
⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑦 𝑥 ) |
15 |
14
|
ad2antrr |
⊢ ( ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) ∧ ¬ ∀ 𝑦 𝑦 = 𝑧 ) → Ⅎ 𝑦 𝑥 ) |
16 |
13 15
|
nfeld |
⊢ ( ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) ∧ ¬ ∀ 𝑦 𝑦 = 𝑧 ) → Ⅎ 𝑦 𝑧 ∈ 𝑥 ) |
17 |
16
|
nf5rd |
⊢ ( ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) ∧ ¬ ∀ 𝑦 𝑦 = 𝑧 ) → ( 𝑧 ∈ 𝑥 → ∀ 𝑦 𝑧 ∈ 𝑥 ) ) |
18 |
|
sp |
⊢ ( ∀ 𝑦 𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑥 ) |
19 |
17 18
|
impbid1 |
⊢ ( ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) ∧ ¬ ∀ 𝑦 𝑦 = 𝑧 ) → ( 𝑧 ∈ 𝑥 ↔ ∀ 𝑦 𝑧 ∈ 𝑥 ) ) |
20 |
|
nfcvf2 |
⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑧 → Ⅎ 𝑧 𝑥 ) |
21 |
20
|
ad2antlr |
⊢ ( ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) ∧ ¬ ∀ 𝑦 𝑦 = 𝑧 ) → Ⅎ 𝑧 𝑥 ) |
22 |
|
nfcvf2 |
⊢ ( ¬ ∀ 𝑦 𝑦 = 𝑧 → Ⅎ 𝑧 𝑦 ) |
23 |
22
|
adantl |
⊢ ( ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) ∧ ¬ ∀ 𝑦 𝑦 = 𝑧 ) → Ⅎ 𝑧 𝑦 ) |
24 |
21 23
|
nfeld |
⊢ ( ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) ∧ ¬ ∀ 𝑦 𝑦 = 𝑧 ) → Ⅎ 𝑧 𝑥 ∈ 𝑦 ) |
25 |
24
|
nf5rd |
⊢ ( ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) ∧ ¬ ∀ 𝑦 𝑦 = 𝑧 ) → ( 𝑥 ∈ 𝑦 → ∀ 𝑧 𝑥 ∈ 𝑦 ) ) |
26 |
|
sp |
⊢ ( ∀ 𝑧 𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑦 ) |
27 |
25 26
|
impbid1 |
⊢ ( ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) ∧ ¬ ∀ 𝑦 𝑦 = 𝑧 ) → ( 𝑥 ∈ 𝑦 ↔ ∀ 𝑧 𝑥 ∈ 𝑦 ) ) |
28 |
27
|
anbi1d |
⊢ ( ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) ∧ ¬ ∀ 𝑦 𝑦 = 𝑧 ) → ( ( 𝑥 ∈ 𝑦 ∧ ∀ 𝑦 𝜑 ) ↔ ( ∀ 𝑧 𝑥 ∈ 𝑦 ∧ ∀ 𝑦 𝜑 ) ) ) |
29 |
6 28
|
exbid |
⊢ ( ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) ∧ ¬ ∀ 𝑦 𝑦 = 𝑧 ) → ( ∃ 𝑥 ( 𝑥 ∈ 𝑦 ∧ ∀ 𝑦 𝜑 ) ↔ ∃ 𝑥 ( ∀ 𝑧 𝑥 ∈ 𝑦 ∧ ∀ 𝑦 𝜑 ) ) ) |
30 |
19 29
|
bibi12d |
⊢ ( ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) ∧ ¬ ∀ 𝑦 𝑦 = 𝑧 ) → ( ( 𝑧 ∈ 𝑥 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝑦 ∧ ∀ 𝑦 𝜑 ) ) ↔ ( ∀ 𝑦 𝑧 ∈ 𝑥 ↔ ∃ 𝑥 ( ∀ 𝑧 𝑥 ∈ 𝑦 ∧ ∀ 𝑦 𝜑 ) ) ) ) |
31 |
11 30
|
albid |
⊢ ( ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) ∧ ¬ ∀ 𝑦 𝑦 = 𝑧 ) → ( ∀ 𝑧 ( 𝑧 ∈ 𝑥 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝑦 ∧ ∀ 𝑦 𝜑 ) ) ↔ ∀ 𝑧 ( ∀ 𝑦 𝑧 ∈ 𝑥 ↔ ∃ 𝑥 ( ∀ 𝑧 𝑥 ∈ 𝑦 ∧ ∀ 𝑦 𝜑 ) ) ) ) |
32 |
31
|
imbi2d |
⊢ ( ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) ∧ ¬ ∀ 𝑦 𝑦 = 𝑧 ) → ( ( ∃ 𝑦 ∀ 𝑧 ( 𝜑 → 𝑧 = 𝑦 ) → ∀ 𝑧 ( 𝑧 ∈ 𝑥 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝑦 ∧ ∀ 𝑦 𝜑 ) ) ) ↔ ( ∃ 𝑦 ∀ 𝑧 ( 𝜑 → 𝑧 = 𝑦 ) → ∀ 𝑧 ( ∀ 𝑦 𝑧 ∈ 𝑥 ↔ ∃ 𝑥 ( ∀ 𝑧 𝑥 ∈ 𝑦 ∧ ∀ 𝑦 𝜑 ) ) ) ) ) |
33 |
6 32
|
exbid |
⊢ ( ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) ∧ ¬ ∀ 𝑦 𝑦 = 𝑧 ) → ( ∃ 𝑥 ( ∃ 𝑦 ∀ 𝑧 ( 𝜑 → 𝑧 = 𝑦 ) → ∀ 𝑧 ( 𝑧 ∈ 𝑥 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝑦 ∧ ∀ 𝑦 𝜑 ) ) ) ↔ ∃ 𝑥 ( ∃ 𝑦 ∀ 𝑧 ( 𝜑 → 𝑧 = 𝑦 ) → ∀ 𝑧 ( ∀ 𝑦 𝑧 ∈ 𝑥 ↔ ∃ 𝑥 ( ∀ 𝑧 𝑥 ∈ 𝑦 ∧ ∀ 𝑦 𝜑 ) ) ) ) ) |
34 |
1 33
|
mpbid |
⊢ ( ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) ∧ ¬ ∀ 𝑦 𝑦 = 𝑧 ) → ∃ 𝑥 ( ∃ 𝑦 ∀ 𝑧 ( 𝜑 → 𝑧 = 𝑦 ) → ∀ 𝑧 ( ∀ 𝑦 𝑧 ∈ 𝑥 ↔ ∃ 𝑥 ( ∀ 𝑧 𝑥 ∈ 𝑦 ∧ ∀ 𝑦 𝜑 ) ) ) ) |
35 |
34
|
exp31 |
⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( ¬ ∀ 𝑥 𝑥 = 𝑧 → ( ¬ ∀ 𝑦 𝑦 = 𝑧 → ∃ 𝑥 ( ∃ 𝑦 ∀ 𝑧 ( 𝜑 → 𝑧 = 𝑦 ) → ∀ 𝑧 ( ∀ 𝑦 𝑧 ∈ 𝑥 ↔ ∃ 𝑥 ( ∀ 𝑧 𝑥 ∈ 𝑦 ∧ ∀ 𝑦 𝜑 ) ) ) ) ) ) |
36 |
|
nfae |
⊢ Ⅎ 𝑧 ∀ 𝑥 𝑥 = 𝑦 |
37 |
|
nd2 |
⊢ ( ∀ 𝑦 𝑦 = 𝑥 → ¬ ∀ 𝑦 𝑧 ∈ 𝑥 ) |
38 |
37
|
aecoms |
⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ¬ ∀ 𝑦 𝑧 ∈ 𝑥 ) |
39 |
|
nfae |
⊢ Ⅎ 𝑥 ∀ 𝑥 𝑥 = 𝑦 |
40 |
|
nd3 |
⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ¬ ∀ 𝑧 𝑥 ∈ 𝑦 ) |
41 |
40
|
intnanrd |
⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ¬ ( ∀ 𝑧 𝑥 ∈ 𝑦 ∧ ∀ 𝑦 𝜑 ) ) |
42 |
39 41
|
nexd |
⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ¬ ∃ 𝑥 ( ∀ 𝑧 𝑥 ∈ 𝑦 ∧ ∀ 𝑦 𝜑 ) ) |
43 |
38 42
|
2falsed |
⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑦 𝑧 ∈ 𝑥 ↔ ∃ 𝑥 ( ∀ 𝑧 𝑥 ∈ 𝑦 ∧ ∀ 𝑦 𝜑 ) ) ) |
44 |
36 43
|
alrimi |
⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ∀ 𝑧 ( ∀ 𝑦 𝑧 ∈ 𝑥 ↔ ∃ 𝑥 ( ∀ 𝑧 𝑥 ∈ 𝑦 ∧ ∀ 𝑦 𝜑 ) ) ) |
45 |
44
|
a1d |
⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∃ 𝑦 ∀ 𝑧 ( 𝜑 → 𝑧 = 𝑦 ) → ∀ 𝑧 ( ∀ 𝑦 𝑧 ∈ 𝑥 ↔ ∃ 𝑥 ( ∀ 𝑧 𝑥 ∈ 𝑦 ∧ ∀ 𝑦 𝜑 ) ) ) ) |
46 |
45
|
19.8ad |
⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ∃ 𝑥 ( ∃ 𝑦 ∀ 𝑧 ( 𝜑 → 𝑧 = 𝑦 ) → ∀ 𝑧 ( ∀ 𝑦 𝑧 ∈ 𝑥 ↔ ∃ 𝑥 ( ∀ 𝑧 𝑥 ∈ 𝑦 ∧ ∀ 𝑦 𝜑 ) ) ) ) |
47 |
|
nfae |
⊢ Ⅎ 𝑧 ∀ 𝑥 𝑥 = 𝑧 |
48 |
|
nd4 |
⊢ ( ∀ 𝑥 𝑥 = 𝑧 → ¬ ∀ 𝑦 𝑧 ∈ 𝑥 ) |
49 |
|
nfae |
⊢ Ⅎ 𝑥 ∀ 𝑥 𝑥 = 𝑧 |
50 |
|
nd1 |
⊢ ( ∀ 𝑧 𝑧 = 𝑥 → ¬ ∀ 𝑧 𝑥 ∈ 𝑦 ) |
51 |
50
|
aecoms |
⊢ ( ∀ 𝑥 𝑥 = 𝑧 → ¬ ∀ 𝑧 𝑥 ∈ 𝑦 ) |
52 |
51
|
intnanrd |
⊢ ( ∀ 𝑥 𝑥 = 𝑧 → ¬ ( ∀ 𝑧 𝑥 ∈ 𝑦 ∧ ∀ 𝑦 𝜑 ) ) |
53 |
49 52
|
nexd |
⊢ ( ∀ 𝑥 𝑥 = 𝑧 → ¬ ∃ 𝑥 ( ∀ 𝑧 𝑥 ∈ 𝑦 ∧ ∀ 𝑦 𝜑 ) ) |
54 |
48 53
|
2falsed |
⊢ ( ∀ 𝑥 𝑥 = 𝑧 → ( ∀ 𝑦 𝑧 ∈ 𝑥 ↔ ∃ 𝑥 ( ∀ 𝑧 𝑥 ∈ 𝑦 ∧ ∀ 𝑦 𝜑 ) ) ) |
55 |
47 54
|
alrimi |
⊢ ( ∀ 𝑥 𝑥 = 𝑧 → ∀ 𝑧 ( ∀ 𝑦 𝑧 ∈ 𝑥 ↔ ∃ 𝑥 ( ∀ 𝑧 𝑥 ∈ 𝑦 ∧ ∀ 𝑦 𝜑 ) ) ) |
56 |
55
|
a1d |
⊢ ( ∀ 𝑥 𝑥 = 𝑧 → ( ∃ 𝑦 ∀ 𝑧 ( 𝜑 → 𝑧 = 𝑦 ) → ∀ 𝑧 ( ∀ 𝑦 𝑧 ∈ 𝑥 ↔ ∃ 𝑥 ( ∀ 𝑧 𝑥 ∈ 𝑦 ∧ ∀ 𝑦 𝜑 ) ) ) ) |
57 |
56
|
19.8ad |
⊢ ( ∀ 𝑥 𝑥 = 𝑧 → ∃ 𝑥 ( ∃ 𝑦 ∀ 𝑧 ( 𝜑 → 𝑧 = 𝑦 ) → ∀ 𝑧 ( ∀ 𝑦 𝑧 ∈ 𝑥 ↔ ∃ 𝑥 ( ∀ 𝑧 𝑥 ∈ 𝑦 ∧ ∀ 𝑦 𝜑 ) ) ) ) |
58 |
|
nfae |
⊢ Ⅎ 𝑧 ∀ 𝑦 𝑦 = 𝑧 |
59 |
|
nd1 |
⊢ ( ∀ 𝑦 𝑦 = 𝑧 → ¬ ∀ 𝑦 𝑧 ∈ 𝑥 ) |
60 |
|
nfae |
⊢ Ⅎ 𝑥 ∀ 𝑦 𝑦 = 𝑧 |
61 |
|
nd2 |
⊢ ( ∀ 𝑧 𝑧 = 𝑦 → ¬ ∀ 𝑧 𝑥 ∈ 𝑦 ) |
62 |
61
|
aecoms |
⊢ ( ∀ 𝑦 𝑦 = 𝑧 → ¬ ∀ 𝑧 𝑥 ∈ 𝑦 ) |
63 |
62
|
intnanrd |
⊢ ( ∀ 𝑦 𝑦 = 𝑧 → ¬ ( ∀ 𝑧 𝑥 ∈ 𝑦 ∧ ∀ 𝑦 𝜑 ) ) |
64 |
60 63
|
nexd |
⊢ ( ∀ 𝑦 𝑦 = 𝑧 → ¬ ∃ 𝑥 ( ∀ 𝑧 𝑥 ∈ 𝑦 ∧ ∀ 𝑦 𝜑 ) ) |
65 |
59 64
|
2falsed |
⊢ ( ∀ 𝑦 𝑦 = 𝑧 → ( ∀ 𝑦 𝑧 ∈ 𝑥 ↔ ∃ 𝑥 ( ∀ 𝑧 𝑥 ∈ 𝑦 ∧ ∀ 𝑦 𝜑 ) ) ) |
66 |
58 65
|
alrimi |
⊢ ( ∀ 𝑦 𝑦 = 𝑧 → ∀ 𝑧 ( ∀ 𝑦 𝑧 ∈ 𝑥 ↔ ∃ 𝑥 ( ∀ 𝑧 𝑥 ∈ 𝑦 ∧ ∀ 𝑦 𝜑 ) ) ) |
67 |
66
|
a1d |
⊢ ( ∀ 𝑦 𝑦 = 𝑧 → ( ∃ 𝑦 ∀ 𝑧 ( 𝜑 → 𝑧 = 𝑦 ) → ∀ 𝑧 ( ∀ 𝑦 𝑧 ∈ 𝑥 ↔ ∃ 𝑥 ( ∀ 𝑧 𝑥 ∈ 𝑦 ∧ ∀ 𝑦 𝜑 ) ) ) ) |
68 |
67
|
19.8ad |
⊢ ( ∀ 𝑦 𝑦 = 𝑧 → ∃ 𝑥 ( ∃ 𝑦 ∀ 𝑧 ( 𝜑 → 𝑧 = 𝑦 ) → ∀ 𝑧 ( ∀ 𝑦 𝑧 ∈ 𝑥 ↔ ∃ 𝑥 ( ∀ 𝑧 𝑥 ∈ 𝑦 ∧ ∀ 𝑦 𝜑 ) ) ) ) |
69 |
35 46 57 68
|
pm2.61iii |
⊢ ∃ 𝑥 ( ∃ 𝑦 ∀ 𝑧 ( 𝜑 → 𝑧 = 𝑦 ) → ∀ 𝑧 ( ∀ 𝑦 𝑧 ∈ 𝑥 ↔ ∃ 𝑥 ( ∀ 𝑧 𝑥 ∈ 𝑦 ∧ ∀ 𝑦 𝜑 ) ) ) |