| Step |
Hyp |
Ref |
Expression |
| 1 |
|
vex |
⊢ 𝑣 ∈ V |
| 2 |
1
|
rabex |
⊢ { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } ∈ V |
| 3 |
|
raleq |
⊢ ( 𝑥 = { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } → ( ∀ 𝑧 ∈ 𝑥 𝑧 ≠ ∅ ↔ ∀ 𝑧 ∈ { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } 𝑧 ≠ ∅ ) ) |
| 4 |
|
raleq |
⊢ ( 𝑥 = { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } → ( ∀ 𝑤 ∈ 𝑥 𝜑 ↔ ∀ 𝑤 ∈ { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } 𝜑 ) ) |
| 5 |
4
|
raleqbi1dv |
⊢ ( 𝑥 = { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } → ( ∀ 𝑧 ∈ 𝑥 ∀ 𝑤 ∈ 𝑥 𝜑 ↔ ∀ 𝑧 ∈ { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } ∀ 𝑤 ∈ { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } 𝜑 ) ) |
| 6 |
3 5
|
anbi12d |
⊢ ( 𝑥 = { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } → ( ( ∀ 𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀ 𝑧 ∈ 𝑥 ∀ 𝑤 ∈ 𝑥 𝜑 ) ↔ ( ∀ 𝑧 ∈ { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } 𝑧 ≠ ∅ ∧ ∀ 𝑧 ∈ { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } ∀ 𝑤 ∈ { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } 𝜑 ) ) ) |
| 7 |
|
raleq |
⊢ ( 𝑥 = { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } → ( ∀ 𝑧 ∈ 𝑥 𝜓 ↔ ∀ 𝑧 ∈ { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } 𝜓 ) ) |
| 8 |
7
|
exbidv |
⊢ ( 𝑥 = { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } → ( ∃ 𝑦 ∀ 𝑧 ∈ 𝑥 𝜓 ↔ ∃ 𝑦 ∀ 𝑧 ∈ { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } 𝜓 ) ) |
| 9 |
6 8
|
imbi12d |
⊢ ( 𝑥 = { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } → ( ( ( ∀ 𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀ 𝑧 ∈ 𝑥 ∀ 𝑤 ∈ 𝑥 𝜑 ) → ∃ 𝑦 ∀ 𝑧 ∈ 𝑥 𝜓 ) ↔ ( ( ∀ 𝑧 ∈ { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } 𝑧 ≠ ∅ ∧ ∀ 𝑧 ∈ { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } ∀ 𝑤 ∈ { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } 𝜑 ) → ∃ 𝑦 ∀ 𝑧 ∈ { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } 𝜓 ) ) ) |
| 10 |
2 9
|
spcv |
⊢ ( ∀ 𝑥 ( ( ∀ 𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀ 𝑧 ∈ 𝑥 ∀ 𝑤 ∈ 𝑥 𝜑 ) → ∃ 𝑦 ∀ 𝑧 ∈ 𝑥 𝜓 ) → ( ( ∀ 𝑧 ∈ { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } 𝑧 ≠ ∅ ∧ ∀ 𝑧 ∈ { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } ∀ 𝑤 ∈ { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } 𝜑 ) → ∃ 𝑦 ∀ 𝑧 ∈ { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } 𝜓 ) ) |
| 11 |
10
|
alrimiv |
⊢ ( ∀ 𝑥 ( ( ∀ 𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀ 𝑧 ∈ 𝑥 ∀ 𝑤 ∈ 𝑥 𝜑 ) → ∃ 𝑦 ∀ 𝑧 ∈ 𝑥 𝜓 ) → ∀ 𝑣 ( ( ∀ 𝑧 ∈ { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } 𝑧 ≠ ∅ ∧ ∀ 𝑧 ∈ { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } ∀ 𝑤 ∈ { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } 𝜑 ) → ∃ 𝑦 ∀ 𝑧 ∈ { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } 𝜓 ) ) |
| 12 |
|
elrabi |
⊢ ( 𝑧 ∈ { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } → 𝑧 ∈ 𝑣 ) |
| 13 |
|
elrabi |
⊢ ( 𝑤 ∈ { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } → 𝑤 ∈ 𝑣 ) |
| 14 |
13
|
imim1i |
⊢ ( ( 𝑤 ∈ 𝑣 → 𝜑 ) → ( 𝑤 ∈ { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } → 𝜑 ) ) |
| 15 |
14
|
ralimi2 |
⊢ ( ∀ 𝑤 ∈ 𝑣 𝜑 → ∀ 𝑤 ∈ { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } 𝜑 ) |
| 16 |
12 15
|
imim12i |
⊢ ( ( 𝑧 ∈ 𝑣 → ∀ 𝑤 ∈ 𝑣 𝜑 ) → ( 𝑧 ∈ { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } → ∀ 𝑤 ∈ { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } 𝜑 ) ) |
| 17 |
16
|
ralimi2 |
⊢ ( ∀ 𝑧 ∈ 𝑣 ∀ 𝑤 ∈ 𝑣 𝜑 → ∀ 𝑧 ∈ { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } ∀ 𝑤 ∈ { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } 𝜑 ) |
| 18 |
|
neeq1 |
⊢ ( 𝑢 = 𝑧 → ( 𝑢 ≠ ∅ ↔ 𝑧 ≠ ∅ ) ) |
| 19 |
18
|
elrab |
⊢ ( 𝑧 ∈ { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } ↔ ( 𝑧 ∈ 𝑣 ∧ 𝑧 ≠ ∅ ) ) |
| 20 |
19
|
simprbi |
⊢ ( 𝑧 ∈ { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } → 𝑧 ≠ ∅ ) |
| 21 |
20
|
rgen |
⊢ ∀ 𝑧 ∈ { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } 𝑧 ≠ ∅ |
| 22 |
17 21
|
jctil |
⊢ ( ∀ 𝑧 ∈ 𝑣 ∀ 𝑤 ∈ 𝑣 𝜑 → ( ∀ 𝑧 ∈ { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } 𝑧 ≠ ∅ ∧ ∀ 𝑧 ∈ { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } ∀ 𝑤 ∈ { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } 𝜑 ) ) |
| 23 |
19
|
biimpri |
⊢ ( ( 𝑧 ∈ 𝑣 ∧ 𝑧 ≠ ∅ ) → 𝑧 ∈ { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } ) |
| 24 |
23
|
imim1i |
⊢ ( ( 𝑧 ∈ { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } → 𝜓 ) → ( ( 𝑧 ∈ 𝑣 ∧ 𝑧 ≠ ∅ ) → 𝜓 ) ) |
| 25 |
24
|
expd |
⊢ ( ( 𝑧 ∈ { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } → 𝜓 ) → ( 𝑧 ∈ 𝑣 → ( 𝑧 ≠ ∅ → 𝜓 ) ) ) |
| 26 |
25
|
ralimi2 |
⊢ ( ∀ 𝑧 ∈ { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } 𝜓 → ∀ 𝑧 ∈ 𝑣 ( 𝑧 ≠ ∅ → 𝜓 ) ) |
| 27 |
26
|
eximi |
⊢ ( ∃ 𝑦 ∀ 𝑧 ∈ { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } 𝜓 → ∃ 𝑦 ∀ 𝑧 ∈ 𝑣 ( 𝑧 ≠ ∅ → 𝜓 ) ) |
| 28 |
22 27
|
imim12i |
⊢ ( ( ( ∀ 𝑧 ∈ { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } 𝑧 ≠ ∅ ∧ ∀ 𝑧 ∈ { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } ∀ 𝑤 ∈ { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } 𝜑 ) → ∃ 𝑦 ∀ 𝑧 ∈ { 𝑢 ∈ 𝑣 ∣ 𝑢 ≠ ∅ } 𝜓 ) → ( ∀ 𝑧 ∈ 𝑣 ∀ 𝑤 ∈ 𝑣 𝜑 → ∃ 𝑦 ∀ 𝑧 ∈ 𝑣 ( 𝑧 ≠ ∅ → 𝜓 ) ) ) |
| 29 |
11 28
|
sylg |
⊢ ( ∀ 𝑥 ( ( ∀ 𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀ 𝑧 ∈ 𝑥 ∀ 𝑤 ∈ 𝑥 𝜑 ) → ∃ 𝑦 ∀ 𝑧 ∈ 𝑥 𝜓 ) → ∀ 𝑣 ( ∀ 𝑧 ∈ 𝑣 ∀ 𝑤 ∈ 𝑣 𝜑 → ∃ 𝑦 ∀ 𝑧 ∈ 𝑣 ( 𝑧 ≠ ∅ → 𝜓 ) ) ) |
| 30 |
|
raleq |
⊢ ( 𝑣 = 𝑥 → ( ∀ 𝑤 ∈ 𝑣 𝜑 ↔ ∀ 𝑤 ∈ 𝑥 𝜑 ) ) |
| 31 |
30
|
raleqbi1dv |
⊢ ( 𝑣 = 𝑥 → ( ∀ 𝑧 ∈ 𝑣 ∀ 𝑤 ∈ 𝑣 𝜑 ↔ ∀ 𝑧 ∈ 𝑥 ∀ 𝑤 ∈ 𝑥 𝜑 ) ) |
| 32 |
|
raleq |
⊢ ( 𝑣 = 𝑥 → ( ∀ 𝑧 ∈ 𝑣 ( 𝑧 ≠ ∅ → 𝜓 ) ↔ ∀ 𝑧 ∈ 𝑥 ( 𝑧 ≠ ∅ → 𝜓 ) ) ) |
| 33 |
32
|
exbidv |
⊢ ( 𝑣 = 𝑥 → ( ∃ 𝑦 ∀ 𝑧 ∈ 𝑣 ( 𝑧 ≠ ∅ → 𝜓 ) ↔ ∃ 𝑦 ∀ 𝑧 ∈ 𝑥 ( 𝑧 ≠ ∅ → 𝜓 ) ) ) |
| 34 |
31 33
|
imbi12d |
⊢ ( 𝑣 = 𝑥 → ( ( ∀ 𝑧 ∈ 𝑣 ∀ 𝑤 ∈ 𝑣 𝜑 → ∃ 𝑦 ∀ 𝑧 ∈ 𝑣 ( 𝑧 ≠ ∅ → 𝜓 ) ) ↔ ( ∀ 𝑧 ∈ 𝑥 ∀ 𝑤 ∈ 𝑥 𝜑 → ∃ 𝑦 ∀ 𝑧 ∈ 𝑥 ( 𝑧 ≠ ∅ → 𝜓 ) ) ) ) |
| 35 |
34
|
cbvalvw |
⊢ ( ∀ 𝑣 ( ∀ 𝑧 ∈ 𝑣 ∀ 𝑤 ∈ 𝑣 𝜑 → ∃ 𝑦 ∀ 𝑧 ∈ 𝑣 ( 𝑧 ≠ ∅ → 𝜓 ) ) ↔ ∀ 𝑥 ( ∀ 𝑧 ∈ 𝑥 ∀ 𝑤 ∈ 𝑥 𝜑 → ∃ 𝑦 ∀ 𝑧 ∈ 𝑥 ( 𝑧 ≠ ∅ → 𝜓 ) ) ) |
| 36 |
29 35
|
sylib |
⊢ ( ∀ 𝑥 ( ( ∀ 𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀ 𝑧 ∈ 𝑥 ∀ 𝑤 ∈ 𝑥 𝜑 ) → ∃ 𝑦 ∀ 𝑧 ∈ 𝑥 𝜓 ) → ∀ 𝑥 ( ∀ 𝑧 ∈ 𝑥 ∀ 𝑤 ∈ 𝑥 𝜑 → ∃ 𝑦 ∀ 𝑧 ∈ 𝑥 ( 𝑧 ≠ ∅ → 𝜓 ) ) ) |