| Step |
Hyp |
Ref |
Expression |
| 1 |
|
eu6w.x |
⊢ ( 𝑥 = 𝑧 → ( 𝜑 ↔ 𝜓 ) ) |
| 2 |
|
eu6w.y |
⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜃 ) ) |
| 3 |
|
alnex |
⊢ ( ∀ 𝑥 ¬ 𝜑 ↔ ¬ ∃ 𝑥 𝜑 ) |
| 4 |
|
pm2.21 |
⊢ ( ¬ 𝜑 → ( 𝜑 → 𝑥 = 𝑦 ) ) |
| 5 |
4
|
alimi |
⊢ ( ∀ 𝑥 ¬ 𝜑 → ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) ) |
| 6 |
3 5
|
sylbir |
⊢ ( ¬ ∃ 𝑥 𝜑 → ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) ) |
| 7 |
|
equequ2 |
⊢ ( 𝑦 = 𝑧 → ( 𝑥 = 𝑦 ↔ 𝑥 = 𝑧 ) ) |
| 8 |
7
|
imbi2d |
⊢ ( 𝑦 = 𝑧 → ( ( 𝜑 → 𝑥 = 𝑦 ) ↔ ( 𝜑 → 𝑥 = 𝑧 ) ) ) |
| 9 |
8
|
albidv |
⊢ ( 𝑦 = 𝑧 → ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) ↔ ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑧 ) ) ) |
| 10 |
9
|
19.8aw |
⊢ ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) → ∃ 𝑦 ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) ) |
| 11 |
6 10
|
syl |
⊢ ( ¬ ∃ 𝑥 𝜑 → ∃ 𝑦 ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) ) |
| 12 |
|
biimp |
⊢ ( ( 𝜑 ↔ 𝑥 = 𝑦 ) → ( 𝜑 → 𝑥 = 𝑦 ) ) |
| 13 |
12
|
alimi |
⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) → ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) ) |
| 14 |
13
|
eximi |
⊢ ( ∃ 𝑦 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) → ∃ 𝑦 ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) ) |
| 15 |
11 14
|
ja |
⊢ ( ( ∃ 𝑥 𝜑 → ∃ 𝑦 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) → ∃ 𝑦 ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) ) |
| 16 |
|
equequ1 |
⊢ ( 𝑥 = 𝑧 → ( 𝑥 = 𝑦 ↔ 𝑧 = 𝑦 ) ) |
| 17 |
1 16
|
imbi12d |
⊢ ( 𝑥 = 𝑧 → ( ( 𝜑 → 𝑥 = 𝑦 ) ↔ ( 𝜓 → 𝑧 = 𝑦 ) ) ) |
| 18 |
17
|
nfa1w |
⊢ Ⅎ 𝑥 ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) |
| 19 |
1 16
|
bibi12d |
⊢ ( 𝑥 = 𝑧 → ( ( 𝜑 ↔ 𝑥 = 𝑦 ) ↔ ( 𝜓 ↔ 𝑧 = 𝑦 ) ) ) |
| 20 |
19
|
nfa1w |
⊢ Ⅎ 𝑥 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) |
| 21 |
18 20
|
nfim |
⊢ Ⅎ 𝑥 ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) → ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) |
| 22 |
|
19.38b |
⊢ ( Ⅎ 𝑥 ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) → ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) → ( ( ∃ 𝑥 𝜑 → ∀ 𝑥 ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) → ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) ) ↔ ∀ 𝑥 ( 𝜑 → ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) → ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) ) ) ) |
| 23 |
17
|
cbvalvw |
⊢ ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) ↔ ∀ 𝑧 ( 𝜓 → 𝑧 = 𝑦 ) ) |
| 24 |
19
|
cbvalvw |
⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ↔ ∀ 𝑧 ( 𝜓 ↔ 𝑧 = 𝑦 ) ) |
| 25 |
23 24
|
imbi12i |
⊢ ( ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) → ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) ↔ ( ∀ 𝑧 ( 𝜓 → 𝑧 = 𝑦 ) → ∀ 𝑧 ( 𝜓 ↔ 𝑧 = 𝑦 ) ) ) |
| 26 |
25
|
a1i |
⊢ ( 𝑥 = 𝑧 → ( ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) → ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) ↔ ( ∀ 𝑧 ( 𝜓 → 𝑧 = 𝑦 ) → ∀ 𝑧 ( 𝜓 ↔ 𝑧 = 𝑦 ) ) ) ) |
| 27 |
26
|
spw |
⊢ ( ∀ 𝑥 ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) → ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) → ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) → ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) ) |
| 28 |
26
|
19.8aw |
⊢ ( ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) → ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) → ∃ 𝑥 ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) → ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) ) |
| 29 |
|
id |
⊢ ( Ⅎ 𝑥 ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) → ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) → Ⅎ 𝑥 ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) → ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) ) |
| 30 |
29
|
nfrd |
⊢ ( Ⅎ 𝑥 ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) → ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) → ( ∃ 𝑥 ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) → ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) → ∀ 𝑥 ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) → ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) ) ) |
| 31 |
28 30
|
syl5 |
⊢ ( Ⅎ 𝑥 ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) → ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) → ( ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) → ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) → ∀ 𝑥 ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) → ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) ) ) |
| 32 |
27 31
|
impbid2 |
⊢ ( Ⅎ 𝑥 ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) → ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) → ( ∀ 𝑥 ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) → ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) ↔ ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) → ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) ) ) |
| 33 |
32
|
imbi2d |
⊢ ( Ⅎ 𝑥 ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) → ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) → ( ( ∃ 𝑥 𝜑 → ∀ 𝑥 ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) → ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) ) ↔ ( ∃ 𝑥 𝜑 → ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) → ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) ) ) ) |
| 34 |
22 33
|
bitr3d |
⊢ ( Ⅎ 𝑥 ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) → ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) → ( ∀ 𝑥 ( 𝜑 → ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) → ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) ) ↔ ( ∃ 𝑥 𝜑 → ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) → ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) ) ) ) |
| 35 |
21 34
|
ax-mp |
⊢ ( ∀ 𝑥 ( 𝜑 → ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) → ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) ) ↔ ( ∃ 𝑥 𝜑 → ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) → ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) ) ) |
| 36 |
17
|
spw |
⊢ ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) → ( 𝜑 → 𝑥 = 𝑦 ) ) |
| 37 |
|
id |
⊢ ( 𝜑 → 𝜑 ) |
| 38 |
2
|
ax12wlem |
⊢ ( 𝑥 = 𝑦 → ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) |
| 39 |
38
|
com12 |
⊢ ( 𝜑 → ( 𝑥 = 𝑦 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) |
| 40 |
37 39
|
embantd |
⊢ ( 𝜑 → ( ( 𝜑 → 𝑥 = 𝑦 ) → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) |
| 41 |
36 40
|
syl5 |
⊢ ( 𝜑 → ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) |
| 42 |
41
|
ancld |
⊢ ( 𝜑 → ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) → ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) ∧ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) ) |
| 43 |
|
albiim |
⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ↔ ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) ∧ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) |
| 44 |
42 43
|
imbitrrdi |
⊢ ( 𝜑 → ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) → ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) ) |
| 45 |
35 44
|
mpgbi |
⊢ ( ∃ 𝑥 𝜑 → ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) → ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) ) |
| 46 |
45
|
eximdv |
⊢ ( ∃ 𝑥 𝜑 → ( ∃ 𝑦 ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) → ∃ 𝑦 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) ) |
| 47 |
46
|
com12 |
⊢ ( ∃ 𝑦 ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) → ( ∃ 𝑥 𝜑 → ∃ 𝑦 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) ) |
| 48 |
15 47
|
impbii |
⊢ ( ( ∃ 𝑥 𝜑 → ∃ 𝑦 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) ↔ ∃ 𝑦 ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) ) |
| 49 |
48
|
anbi2i |
⊢ ( ( ∃ 𝑥 𝜑 ∧ ( ∃ 𝑥 𝜑 → ∃ 𝑦 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) ) ↔ ( ∃ 𝑥 𝜑 ∧ ∃ 𝑦 ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) ) ) |
| 50 |
|
abai |
⊢ ( ( ∃ 𝑥 𝜑 ∧ ∃ 𝑦 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) ↔ ( ∃ 𝑥 𝜑 ∧ ( ∃ 𝑥 𝜑 → ∃ 𝑦 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) ) ) |
| 51 |
|
eu3v |
⊢ ( ∃! 𝑥 𝜑 ↔ ( ∃ 𝑥 𝜑 ∧ ∃ 𝑦 ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) ) ) |
| 52 |
49 50 51
|
3bitr4ri |
⊢ ( ∃! 𝑥 𝜑 ↔ ( ∃ 𝑥 𝜑 ∧ ∃ 𝑦 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) ) |
| 53 |
|
abai |
⊢ ( ( ∃ 𝑦 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ∧ ∃ 𝑥 𝜑 ) ↔ ( ∃ 𝑦 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ∧ ( ∃ 𝑦 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) → ∃ 𝑥 𝜑 ) ) ) |
| 54 |
|
ancom |
⊢ ( ( ∃ 𝑥 𝜑 ∧ ∃ 𝑦 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) ↔ ( ∃ 𝑦 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ∧ ∃ 𝑥 𝜑 ) ) |
| 55 |
|
biimpr |
⊢ ( ( 𝜑 ↔ 𝑥 = 𝑦 ) → ( 𝑥 = 𝑦 → 𝜑 ) ) |
| 56 |
55
|
alimi |
⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) |
| 57 |
56
|
eximi |
⊢ ( ∃ 𝑦 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) → ∃ 𝑦 ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) |
| 58 |
|
exsbim |
⊢ ( ∃ 𝑦 ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → ∃ 𝑥 𝜑 ) |
| 59 |
57 58
|
syl |
⊢ ( ∃ 𝑦 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) → ∃ 𝑥 𝜑 ) |
| 60 |
59
|
biantru |
⊢ ( ∃ 𝑦 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ↔ ( ∃ 𝑦 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ∧ ( ∃ 𝑦 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) → ∃ 𝑥 𝜑 ) ) ) |
| 61 |
53 54 60
|
3bitr4i |
⊢ ( ( ∃ 𝑥 𝜑 ∧ ∃ 𝑦 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) ↔ ∃ 𝑦 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) |
| 62 |
52 61
|
bitri |
⊢ ( ∃! 𝑥 𝜑 ↔ ∃ 𝑦 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) |