Step |
Hyp |
Ref |
Expression |
1 |
|
findcard3.1 |
⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜒 ) ) |
2 |
|
findcard3.2 |
⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜏 ) ) |
3 |
|
findcard3.3 |
⊢ ( 𝑦 ∈ Fin → ( ∀ 𝑥 ( 𝑥 ⊊ 𝑦 → 𝜑 ) → 𝜒 ) ) |
4 |
|
isfi |
⊢ ( 𝐴 ∈ Fin ↔ ∃ 𝑤 ∈ ω 𝐴 ≈ 𝑤 ) |
5 |
|
nnon |
⊢ ( 𝑤 ∈ ω → 𝑤 ∈ On ) |
6 |
|
eleq1w |
⊢ ( 𝑤 = 𝑧 → ( 𝑤 ∈ ω ↔ 𝑧 ∈ ω ) ) |
7 |
|
breq2 |
⊢ ( 𝑤 = 𝑧 → ( 𝑥 ≈ 𝑤 ↔ 𝑥 ≈ 𝑧 ) ) |
8 |
7
|
imbi1d |
⊢ ( 𝑤 = 𝑧 → ( ( 𝑥 ≈ 𝑤 → 𝜑 ) ↔ ( 𝑥 ≈ 𝑧 → 𝜑 ) ) ) |
9 |
8
|
albidv |
⊢ ( 𝑤 = 𝑧 → ( ∀ 𝑥 ( 𝑥 ≈ 𝑤 → 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 ≈ 𝑧 → 𝜑 ) ) ) |
10 |
6 9
|
imbi12d |
⊢ ( 𝑤 = 𝑧 → ( ( 𝑤 ∈ ω → ∀ 𝑥 ( 𝑥 ≈ 𝑤 → 𝜑 ) ) ↔ ( 𝑧 ∈ ω → ∀ 𝑥 ( 𝑥 ≈ 𝑧 → 𝜑 ) ) ) ) |
11 |
|
rspe |
⊢ ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) → ∃ 𝑤 ∈ ω 𝑦 ≈ 𝑤 ) |
12 |
|
isfi |
⊢ ( 𝑦 ∈ Fin ↔ ∃ 𝑤 ∈ ω 𝑦 ≈ 𝑤 ) |
13 |
11 12
|
sylibr |
⊢ ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) → 𝑦 ∈ Fin ) |
14 |
|
19.21v |
⊢ ( ∀ 𝑥 ( 𝑧 ∈ ω → ( 𝑥 ≈ 𝑧 → 𝜑 ) ) ↔ ( 𝑧 ∈ ω → ∀ 𝑥 ( 𝑥 ≈ 𝑧 → 𝜑 ) ) ) |
15 |
14
|
ralbii |
⊢ ( ∀ 𝑧 ∈ 𝑤 ∀ 𝑥 ( 𝑧 ∈ ω → ( 𝑥 ≈ 𝑧 → 𝜑 ) ) ↔ ∀ 𝑧 ∈ 𝑤 ( 𝑧 ∈ ω → ∀ 𝑥 ( 𝑥 ≈ 𝑧 → 𝜑 ) ) ) |
16 |
|
ralcom4 |
⊢ ( ∀ 𝑧 ∈ 𝑤 ∀ 𝑥 ( 𝑧 ∈ ω → ( 𝑥 ≈ 𝑧 → 𝜑 ) ) ↔ ∀ 𝑥 ∀ 𝑧 ∈ 𝑤 ( 𝑧 ∈ ω → ( 𝑥 ≈ 𝑧 → 𝜑 ) ) ) |
17 |
15 16
|
bitr3i |
⊢ ( ∀ 𝑧 ∈ 𝑤 ( 𝑧 ∈ ω → ∀ 𝑥 ( 𝑥 ≈ 𝑧 → 𝜑 ) ) ↔ ∀ 𝑥 ∀ 𝑧 ∈ 𝑤 ( 𝑧 ∈ ω → ( 𝑥 ≈ 𝑧 → 𝜑 ) ) ) |
18 |
|
pssss |
⊢ ( 𝑥 ⊊ 𝑦 → 𝑥 ⊆ 𝑦 ) |
19 |
|
ssfi |
⊢ ( ( 𝑦 ∈ Fin ∧ 𝑥 ⊆ 𝑦 ) → 𝑥 ∈ Fin ) |
20 |
|
isfi |
⊢ ( 𝑥 ∈ Fin ↔ ∃ 𝑧 ∈ ω 𝑥 ≈ 𝑧 ) |
21 |
19 20
|
sylib |
⊢ ( ( 𝑦 ∈ Fin ∧ 𝑥 ⊆ 𝑦 ) → ∃ 𝑧 ∈ ω 𝑥 ≈ 𝑧 ) |
22 |
13 18 21
|
syl2an |
⊢ ( ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) ∧ 𝑥 ⊊ 𝑦 ) → ∃ 𝑧 ∈ ω 𝑥 ≈ 𝑧 ) |
23 |
|
simprl |
⊢ ( ( ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) ∧ 𝑥 ⊊ 𝑦 ) ∧ ( 𝑧 ∈ ω ∧ 𝑥 ≈ 𝑧 ) ) → 𝑧 ∈ ω ) |
24 |
|
nnfi |
⊢ ( 𝑧 ∈ ω → 𝑧 ∈ Fin ) |
25 |
|
ensymfib |
⊢ ( 𝑧 ∈ Fin → ( 𝑧 ≈ 𝑥 ↔ 𝑥 ≈ 𝑧 ) ) |
26 |
24 25
|
syl |
⊢ ( 𝑧 ∈ ω → ( 𝑧 ≈ 𝑥 ↔ 𝑥 ≈ 𝑧 ) ) |
27 |
26
|
biimpar |
⊢ ( ( 𝑧 ∈ ω ∧ 𝑥 ≈ 𝑧 ) → 𝑧 ≈ 𝑥 ) |
28 |
27
|
adantl |
⊢ ( ( ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) ∧ 𝑥 ⊊ 𝑦 ) ∧ ( 𝑧 ∈ ω ∧ 𝑥 ≈ 𝑧 ) ) → 𝑧 ≈ 𝑥 ) |
29 |
|
simplll |
⊢ ( ( ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) ∧ 𝑥 ⊊ 𝑦 ) ∧ ( 𝑧 ∈ ω ∧ 𝑥 ≈ 𝑧 ) ) → 𝑤 ∈ ω ) |
30 |
|
php3 |
⊢ ( ( 𝑦 ∈ Fin ∧ 𝑥 ⊊ 𝑦 ) → 𝑥 ≺ 𝑦 ) |
31 |
13 30
|
sylan |
⊢ ( ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) ∧ 𝑥 ⊊ 𝑦 ) → 𝑥 ≺ 𝑦 ) |
32 |
31
|
adantr |
⊢ ( ( ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) ∧ 𝑥 ⊊ 𝑦 ) ∧ ( 𝑧 ∈ ω ∧ 𝑥 ≈ 𝑧 ) ) → 𝑥 ≺ 𝑦 ) |
33 |
|
simpllr |
⊢ ( ( ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) ∧ 𝑥 ⊊ 𝑦 ) ∧ ( 𝑧 ∈ ω ∧ 𝑥 ≈ 𝑧 ) ) → 𝑦 ≈ 𝑤 ) |
34 |
|
endom |
⊢ ( 𝑦 ≈ 𝑤 → 𝑦 ≼ 𝑤 ) |
35 |
|
nnfi |
⊢ ( 𝑤 ∈ ω → 𝑤 ∈ Fin ) |
36 |
|
domfi |
⊢ ( ( 𝑤 ∈ Fin ∧ 𝑦 ≼ 𝑤 ) → 𝑦 ∈ Fin ) |
37 |
35 36
|
sylan |
⊢ ( ( 𝑤 ∈ ω ∧ 𝑦 ≼ 𝑤 ) → 𝑦 ∈ Fin ) |
38 |
37
|
3adant2 |
⊢ ( ( 𝑤 ∈ ω ∧ 𝑥 ≺ 𝑦 ∧ 𝑦 ≼ 𝑤 ) → 𝑦 ∈ Fin ) |
39 |
|
sdomdom |
⊢ ( 𝑥 ≺ 𝑦 → 𝑥 ≼ 𝑦 ) |
40 |
|
domfi |
⊢ ( ( 𝑦 ∈ Fin ∧ 𝑥 ≼ 𝑦 ) → 𝑥 ∈ Fin ) |
41 |
39 40
|
sylan2 |
⊢ ( ( 𝑦 ∈ Fin ∧ 𝑥 ≺ 𝑦 ) → 𝑥 ∈ Fin ) |
42 |
41
|
3adant3 |
⊢ ( ( 𝑦 ∈ Fin ∧ 𝑥 ≺ 𝑦 ∧ 𝑦 ≼ 𝑤 ) → 𝑥 ∈ Fin ) |
43 |
38 42
|
syld3an1 |
⊢ ( ( 𝑤 ∈ ω ∧ 𝑥 ≺ 𝑦 ∧ 𝑦 ≼ 𝑤 ) → 𝑥 ∈ Fin ) |
44 |
|
sdomdomtrfi |
⊢ ( ( 𝑥 ∈ Fin ∧ 𝑥 ≺ 𝑦 ∧ 𝑦 ≼ 𝑤 ) → 𝑥 ≺ 𝑤 ) |
45 |
43 44
|
syld3an1 |
⊢ ( ( 𝑤 ∈ ω ∧ 𝑥 ≺ 𝑦 ∧ 𝑦 ≼ 𝑤 ) → 𝑥 ≺ 𝑤 ) |
46 |
34 45
|
syl3an3 |
⊢ ( ( 𝑤 ∈ ω ∧ 𝑥 ≺ 𝑦 ∧ 𝑦 ≈ 𝑤 ) → 𝑥 ≺ 𝑤 ) |
47 |
29 32 33 46
|
syl3anc |
⊢ ( ( ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) ∧ 𝑥 ⊊ 𝑦 ) ∧ ( 𝑧 ∈ ω ∧ 𝑥 ≈ 𝑧 ) ) → 𝑥 ≺ 𝑤 ) |
48 |
|
endom |
⊢ ( 𝑧 ≈ 𝑥 → 𝑧 ≼ 𝑥 ) |
49 |
|
domsdomtrfi |
⊢ ( ( 𝑧 ∈ Fin ∧ 𝑧 ≼ 𝑥 ∧ 𝑥 ≺ 𝑤 ) → 𝑧 ≺ 𝑤 ) |
50 |
24 49
|
syl3an1 |
⊢ ( ( 𝑧 ∈ ω ∧ 𝑧 ≼ 𝑥 ∧ 𝑥 ≺ 𝑤 ) → 𝑧 ≺ 𝑤 ) |
51 |
48 50
|
syl3an2 |
⊢ ( ( 𝑧 ∈ ω ∧ 𝑧 ≈ 𝑥 ∧ 𝑥 ≺ 𝑤 ) → 𝑧 ≺ 𝑤 ) |
52 |
23 28 47 51
|
syl3anc |
⊢ ( ( ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) ∧ 𝑥 ⊊ 𝑦 ) ∧ ( 𝑧 ∈ ω ∧ 𝑥 ≈ 𝑧 ) ) → 𝑧 ≺ 𝑤 ) |
53 |
|
nnsdomo |
⊢ ( ( 𝑧 ∈ ω ∧ 𝑤 ∈ ω ) → ( 𝑧 ≺ 𝑤 ↔ 𝑧 ⊊ 𝑤 ) ) |
54 |
|
nnord |
⊢ ( 𝑧 ∈ ω → Ord 𝑧 ) |
55 |
|
nnord |
⊢ ( 𝑤 ∈ ω → Ord 𝑤 ) |
56 |
|
ordelpss |
⊢ ( ( Ord 𝑧 ∧ Ord 𝑤 ) → ( 𝑧 ∈ 𝑤 ↔ 𝑧 ⊊ 𝑤 ) ) |
57 |
54 55 56
|
syl2an |
⊢ ( ( 𝑧 ∈ ω ∧ 𝑤 ∈ ω ) → ( 𝑧 ∈ 𝑤 ↔ 𝑧 ⊊ 𝑤 ) ) |
58 |
53 57
|
bitr4d |
⊢ ( ( 𝑧 ∈ ω ∧ 𝑤 ∈ ω ) → ( 𝑧 ≺ 𝑤 ↔ 𝑧 ∈ 𝑤 ) ) |
59 |
23 29 58
|
syl2anc |
⊢ ( ( ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) ∧ 𝑥 ⊊ 𝑦 ) ∧ ( 𝑧 ∈ ω ∧ 𝑥 ≈ 𝑧 ) ) → ( 𝑧 ≺ 𝑤 ↔ 𝑧 ∈ 𝑤 ) ) |
60 |
52 59
|
mpbid |
⊢ ( ( ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) ∧ 𝑥 ⊊ 𝑦 ) ∧ ( 𝑧 ∈ ω ∧ 𝑥 ≈ 𝑧 ) ) → 𝑧 ∈ 𝑤 ) |
61 |
60
|
ex |
⊢ ( ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) ∧ 𝑥 ⊊ 𝑦 ) → ( ( 𝑧 ∈ ω ∧ 𝑥 ≈ 𝑧 ) → 𝑧 ∈ 𝑤 ) ) |
62 |
|
simpr |
⊢ ( ( 𝑧 ∈ ω ∧ 𝑥 ≈ 𝑧 ) → 𝑥 ≈ 𝑧 ) |
63 |
61 62
|
jca2 |
⊢ ( ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) ∧ 𝑥 ⊊ 𝑦 ) → ( ( 𝑧 ∈ ω ∧ 𝑥 ≈ 𝑧 ) → ( 𝑧 ∈ 𝑤 ∧ 𝑥 ≈ 𝑧 ) ) ) |
64 |
63
|
reximdv2 |
⊢ ( ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) ∧ 𝑥 ⊊ 𝑦 ) → ( ∃ 𝑧 ∈ ω 𝑥 ≈ 𝑧 → ∃ 𝑧 ∈ 𝑤 𝑥 ≈ 𝑧 ) ) |
65 |
22 64
|
mpd |
⊢ ( ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) ∧ 𝑥 ⊊ 𝑦 ) → ∃ 𝑧 ∈ 𝑤 𝑥 ≈ 𝑧 ) |
66 |
|
r19.29 |
⊢ ( ( ∀ 𝑧 ∈ 𝑤 ( 𝑧 ∈ ω → ( 𝑥 ≈ 𝑧 → 𝜑 ) ) ∧ ∃ 𝑧 ∈ 𝑤 𝑥 ≈ 𝑧 ) → ∃ 𝑧 ∈ 𝑤 ( ( 𝑧 ∈ ω → ( 𝑥 ≈ 𝑧 → 𝜑 ) ) ∧ 𝑥 ≈ 𝑧 ) ) |
67 |
66
|
expcom |
⊢ ( ∃ 𝑧 ∈ 𝑤 𝑥 ≈ 𝑧 → ( ∀ 𝑧 ∈ 𝑤 ( 𝑧 ∈ ω → ( 𝑥 ≈ 𝑧 → 𝜑 ) ) → ∃ 𝑧 ∈ 𝑤 ( ( 𝑧 ∈ ω → ( 𝑥 ≈ 𝑧 → 𝜑 ) ) ∧ 𝑥 ≈ 𝑧 ) ) ) |
68 |
65 67
|
syl |
⊢ ( ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) ∧ 𝑥 ⊊ 𝑦 ) → ( ∀ 𝑧 ∈ 𝑤 ( 𝑧 ∈ ω → ( 𝑥 ≈ 𝑧 → 𝜑 ) ) → ∃ 𝑧 ∈ 𝑤 ( ( 𝑧 ∈ ω → ( 𝑥 ≈ 𝑧 → 𝜑 ) ) ∧ 𝑥 ≈ 𝑧 ) ) ) |
69 |
|
ordom |
⊢ Ord ω |
70 |
|
ordelss |
⊢ ( ( Ord ω ∧ 𝑤 ∈ ω ) → 𝑤 ⊆ ω ) |
71 |
69 70
|
mpan |
⊢ ( 𝑤 ∈ ω → 𝑤 ⊆ ω ) |
72 |
71
|
ad2antrr |
⊢ ( ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) ∧ 𝑥 ⊊ 𝑦 ) → 𝑤 ⊆ ω ) |
73 |
72
|
sseld |
⊢ ( ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) ∧ 𝑥 ⊊ 𝑦 ) → ( 𝑧 ∈ 𝑤 → 𝑧 ∈ ω ) ) |
74 |
|
pm2.27 |
⊢ ( 𝑧 ∈ ω → ( ( 𝑧 ∈ ω → ( 𝑥 ≈ 𝑧 → 𝜑 ) ) → ( 𝑥 ≈ 𝑧 → 𝜑 ) ) ) |
75 |
74
|
impd |
⊢ ( 𝑧 ∈ ω → ( ( ( 𝑧 ∈ ω → ( 𝑥 ≈ 𝑧 → 𝜑 ) ) ∧ 𝑥 ≈ 𝑧 ) → 𝜑 ) ) |
76 |
73 75
|
syl6 |
⊢ ( ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) ∧ 𝑥 ⊊ 𝑦 ) → ( 𝑧 ∈ 𝑤 → ( ( ( 𝑧 ∈ ω → ( 𝑥 ≈ 𝑧 → 𝜑 ) ) ∧ 𝑥 ≈ 𝑧 ) → 𝜑 ) ) ) |
77 |
76
|
rexlimdv |
⊢ ( ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) ∧ 𝑥 ⊊ 𝑦 ) → ( ∃ 𝑧 ∈ 𝑤 ( ( 𝑧 ∈ ω → ( 𝑥 ≈ 𝑧 → 𝜑 ) ) ∧ 𝑥 ≈ 𝑧 ) → 𝜑 ) ) |
78 |
68 77
|
syld |
⊢ ( ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) ∧ 𝑥 ⊊ 𝑦 ) → ( ∀ 𝑧 ∈ 𝑤 ( 𝑧 ∈ ω → ( 𝑥 ≈ 𝑧 → 𝜑 ) ) → 𝜑 ) ) |
79 |
78
|
ex |
⊢ ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) → ( 𝑥 ⊊ 𝑦 → ( ∀ 𝑧 ∈ 𝑤 ( 𝑧 ∈ ω → ( 𝑥 ≈ 𝑧 → 𝜑 ) ) → 𝜑 ) ) ) |
80 |
79
|
com23 |
⊢ ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) → ( ∀ 𝑧 ∈ 𝑤 ( 𝑧 ∈ ω → ( 𝑥 ≈ 𝑧 → 𝜑 ) ) → ( 𝑥 ⊊ 𝑦 → 𝜑 ) ) ) |
81 |
80
|
alimdv |
⊢ ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) → ( ∀ 𝑥 ∀ 𝑧 ∈ 𝑤 ( 𝑧 ∈ ω → ( 𝑥 ≈ 𝑧 → 𝜑 ) ) → ∀ 𝑥 ( 𝑥 ⊊ 𝑦 → 𝜑 ) ) ) |
82 |
17 81
|
biimtrid |
⊢ ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) → ( ∀ 𝑧 ∈ 𝑤 ( 𝑧 ∈ ω → ∀ 𝑥 ( 𝑥 ≈ 𝑧 → 𝜑 ) ) → ∀ 𝑥 ( 𝑥 ⊊ 𝑦 → 𝜑 ) ) ) |
83 |
13 82 3
|
sylsyld |
⊢ ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) → ( ∀ 𝑧 ∈ 𝑤 ( 𝑧 ∈ ω → ∀ 𝑥 ( 𝑥 ≈ 𝑧 → 𝜑 ) ) → 𝜒 ) ) |
84 |
83
|
impancom |
⊢ ( ( 𝑤 ∈ ω ∧ ∀ 𝑧 ∈ 𝑤 ( 𝑧 ∈ ω → ∀ 𝑥 ( 𝑥 ≈ 𝑧 → 𝜑 ) ) ) → ( 𝑦 ≈ 𝑤 → 𝜒 ) ) |
85 |
84
|
alrimiv |
⊢ ( ( 𝑤 ∈ ω ∧ ∀ 𝑧 ∈ 𝑤 ( 𝑧 ∈ ω → ∀ 𝑥 ( 𝑥 ≈ 𝑧 → 𝜑 ) ) ) → ∀ 𝑦 ( 𝑦 ≈ 𝑤 → 𝜒 ) ) |
86 |
85
|
expcom |
⊢ ( ∀ 𝑧 ∈ 𝑤 ( 𝑧 ∈ ω → ∀ 𝑥 ( 𝑥 ≈ 𝑧 → 𝜑 ) ) → ( 𝑤 ∈ ω → ∀ 𝑦 ( 𝑦 ≈ 𝑤 → 𝜒 ) ) ) |
87 |
|
breq1 |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 ≈ 𝑤 ↔ 𝑦 ≈ 𝑤 ) ) |
88 |
87 1
|
imbi12d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 ≈ 𝑤 → 𝜑 ) ↔ ( 𝑦 ≈ 𝑤 → 𝜒 ) ) ) |
89 |
88
|
cbvalvw |
⊢ ( ∀ 𝑥 ( 𝑥 ≈ 𝑤 → 𝜑 ) ↔ ∀ 𝑦 ( 𝑦 ≈ 𝑤 → 𝜒 ) ) |
90 |
86 89
|
imbitrrdi |
⊢ ( ∀ 𝑧 ∈ 𝑤 ( 𝑧 ∈ ω → ∀ 𝑥 ( 𝑥 ≈ 𝑧 → 𝜑 ) ) → ( 𝑤 ∈ ω → ∀ 𝑥 ( 𝑥 ≈ 𝑤 → 𝜑 ) ) ) |
91 |
90
|
a1i |
⊢ ( 𝑤 ∈ On → ( ∀ 𝑧 ∈ 𝑤 ( 𝑧 ∈ ω → ∀ 𝑥 ( 𝑥 ≈ 𝑧 → 𝜑 ) ) → ( 𝑤 ∈ ω → ∀ 𝑥 ( 𝑥 ≈ 𝑤 → 𝜑 ) ) ) ) |
92 |
10 91
|
tfis2 |
⊢ ( 𝑤 ∈ On → ( 𝑤 ∈ ω → ∀ 𝑥 ( 𝑥 ≈ 𝑤 → 𝜑 ) ) ) |
93 |
5 92
|
mpcom |
⊢ ( 𝑤 ∈ ω → ∀ 𝑥 ( 𝑥 ≈ 𝑤 → 𝜑 ) ) |
94 |
93
|
rgen |
⊢ ∀ 𝑤 ∈ ω ∀ 𝑥 ( 𝑥 ≈ 𝑤 → 𝜑 ) |
95 |
|
r19.29 |
⊢ ( ( ∀ 𝑤 ∈ ω ∀ 𝑥 ( 𝑥 ≈ 𝑤 → 𝜑 ) ∧ ∃ 𝑤 ∈ ω 𝐴 ≈ 𝑤 ) → ∃ 𝑤 ∈ ω ( ∀ 𝑥 ( 𝑥 ≈ 𝑤 → 𝜑 ) ∧ 𝐴 ≈ 𝑤 ) ) |
96 |
94 95
|
mpan |
⊢ ( ∃ 𝑤 ∈ ω 𝐴 ≈ 𝑤 → ∃ 𝑤 ∈ ω ( ∀ 𝑥 ( 𝑥 ≈ 𝑤 → 𝜑 ) ∧ 𝐴 ≈ 𝑤 ) ) |
97 |
4 96
|
sylbi |
⊢ ( 𝐴 ∈ Fin → ∃ 𝑤 ∈ ω ( ∀ 𝑥 ( 𝑥 ≈ 𝑤 → 𝜑 ) ∧ 𝐴 ≈ 𝑤 ) ) |
98 |
|
breq1 |
⊢ ( 𝑥 = 𝐴 → ( 𝑥 ≈ 𝑤 ↔ 𝐴 ≈ 𝑤 ) ) |
99 |
98 2
|
imbi12d |
⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 ≈ 𝑤 → 𝜑 ) ↔ ( 𝐴 ≈ 𝑤 → 𝜏 ) ) ) |
100 |
99
|
spcgv |
⊢ ( 𝐴 ∈ Fin → ( ∀ 𝑥 ( 𝑥 ≈ 𝑤 → 𝜑 ) → ( 𝐴 ≈ 𝑤 → 𝜏 ) ) ) |
101 |
100
|
impd |
⊢ ( 𝐴 ∈ Fin → ( ( ∀ 𝑥 ( 𝑥 ≈ 𝑤 → 𝜑 ) ∧ 𝐴 ≈ 𝑤 ) → 𝜏 ) ) |
102 |
101
|
rexlimdvw |
⊢ ( 𝐴 ∈ Fin → ( ∃ 𝑤 ∈ ω ( ∀ 𝑥 ( 𝑥 ≈ 𝑤 → 𝜑 ) ∧ 𝐴 ≈ 𝑤 ) → 𝜏 ) ) |
103 |
97 102
|
mpd |
⊢ ( 𝐴 ∈ Fin → 𝜏 ) |