Step |
Hyp |
Ref |
Expression |
1 |
|
dtru |
⊢ ¬ ∀ 𝑦 𝑦 = 𝑧 |
2 |
|
exnal |
⊢ ( ∃ 𝑦 ¬ 𝑦 = 𝑧 ↔ ¬ ∀ 𝑦 𝑦 = 𝑧 ) |
3 |
1 2
|
mpbir |
⊢ ∃ 𝑦 ¬ 𝑦 = 𝑧 |
4 |
|
nfe1 |
⊢ Ⅎ 𝑦 ∃ 𝑦 ∀ 𝑧 ( ∀ 𝑦 ( ∃ 𝑥 𝑦 ∈ 𝑧 → ∀ 𝑧 𝑦 ∈ 𝑥 ) → 𝑧 ∈ 𝑦 ) |
5 |
|
axpownd |
⊢ ( ¬ 𝑦 = 𝑧 → ∃ 𝑦 ∀ 𝑧 ( ∀ 𝑦 ( ∃ 𝑥 𝑦 ∈ 𝑧 → ∀ 𝑧 𝑦 ∈ 𝑥 ) → 𝑧 ∈ 𝑦 ) ) |
6 |
4 5
|
exlimi |
⊢ ( ∃ 𝑦 ¬ 𝑦 = 𝑧 → ∃ 𝑦 ∀ 𝑧 ( ∀ 𝑦 ( ∃ 𝑥 𝑦 ∈ 𝑧 → ∀ 𝑧 𝑦 ∈ 𝑥 ) → 𝑧 ∈ 𝑦 ) ) |
7 |
3 6
|
ax-mp |
⊢ ∃ 𝑦 ∀ 𝑧 ( ∀ 𝑦 ( ∃ 𝑥 𝑦 ∈ 𝑧 → ∀ 𝑧 𝑦 ∈ 𝑥 ) → 𝑧 ∈ 𝑦 ) |
8 |
|
19.9v |
⊢ ( ∃ 𝑥 𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧 ) |
9 |
|
19.3v |
⊢ ( ∀ 𝑧 𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥 ) |
10 |
8 9
|
imbi12i |
⊢ ( ( ∃ 𝑥 𝑦 ∈ 𝑧 → ∀ 𝑧 𝑦 ∈ 𝑥 ) ↔ ( 𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥 ) ) |
11 |
10
|
albii |
⊢ ( ∀ 𝑦 ( ∃ 𝑥 𝑦 ∈ 𝑧 → ∀ 𝑧 𝑦 ∈ 𝑥 ) ↔ ∀ 𝑦 ( 𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥 ) ) |
12 |
11
|
imbi1i |
⊢ ( ( ∀ 𝑦 ( ∃ 𝑥 𝑦 ∈ 𝑧 → ∀ 𝑧 𝑦 ∈ 𝑥 ) → 𝑧 ∈ 𝑦 ) ↔ ( ∀ 𝑦 ( 𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥 ) → 𝑧 ∈ 𝑦 ) ) |
13 |
12
|
albii |
⊢ ( ∀ 𝑧 ( ∀ 𝑦 ( ∃ 𝑥 𝑦 ∈ 𝑧 → ∀ 𝑧 𝑦 ∈ 𝑥 ) → 𝑧 ∈ 𝑦 ) ↔ ∀ 𝑧 ( ∀ 𝑦 ( 𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥 ) → 𝑧 ∈ 𝑦 ) ) |
14 |
13
|
exbii |
⊢ ( ∃ 𝑦 ∀ 𝑧 ( ∀ 𝑦 ( ∃ 𝑥 𝑦 ∈ 𝑧 → ∀ 𝑧 𝑦 ∈ 𝑥 ) → 𝑧 ∈ 𝑦 ) ↔ ∃ 𝑦 ∀ 𝑧 ( ∀ 𝑦 ( 𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥 ) → 𝑧 ∈ 𝑦 ) ) |
15 |
7 14
|
mpbi |
⊢ ∃ 𝑦 ∀ 𝑧 ( ∀ 𝑦 ( 𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥 ) → 𝑧 ∈ 𝑦 ) |
16 |
|
elequ1 |
⊢ ( 𝑤 = 𝑦 → ( 𝑤 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧 ) ) |
17 |
|
elequ1 |
⊢ ( 𝑤 = 𝑦 → ( 𝑤 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥 ) ) |
18 |
16 17
|
imbi12d |
⊢ ( 𝑤 = 𝑦 → ( ( 𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥 ) ↔ ( 𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥 ) ) ) |
19 |
18
|
cbvalvw |
⊢ ( ∀ 𝑤 ( 𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥 ) ↔ ∀ 𝑦 ( 𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥 ) ) |
20 |
19
|
imbi1i |
⊢ ( ( ∀ 𝑤 ( 𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥 ) → 𝑧 ∈ 𝑦 ) ↔ ( ∀ 𝑦 ( 𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥 ) → 𝑧 ∈ 𝑦 ) ) |
21 |
20
|
albii |
⊢ ( ∀ 𝑧 ( ∀ 𝑤 ( 𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥 ) → 𝑧 ∈ 𝑦 ) ↔ ∀ 𝑧 ( ∀ 𝑦 ( 𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥 ) → 𝑧 ∈ 𝑦 ) ) |
22 |
21
|
exbii |
⊢ ( ∃ 𝑦 ∀ 𝑧 ( ∀ 𝑤 ( 𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥 ) → 𝑧 ∈ 𝑦 ) ↔ ∃ 𝑦 ∀ 𝑧 ( ∀ 𝑦 ( 𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥 ) → 𝑧 ∈ 𝑦 ) ) |
23 |
15 22
|
mpbir |
⊢ ∃ 𝑦 ∀ 𝑧 ( ∀ 𝑤 ( 𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥 ) → 𝑧 ∈ 𝑦 ) |