Step |
Hyp |
Ref |
Expression |
1 |
|
en2lp |
⊢ ¬ ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦 ) |
2 |
|
elequ2 |
⊢ ( 𝑦 = 𝑧 → ( 𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧 ) ) |
3 |
2
|
anbi2d |
⊢ ( 𝑦 = 𝑧 → ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦 ) ↔ ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) ) ) |
4 |
1 3
|
mtbii |
⊢ ( 𝑦 = 𝑧 → ¬ ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) ) |
5 |
4
|
sps |
⊢ ( ∀ 𝑦 𝑦 = 𝑧 → ¬ ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) ) |
6 |
5
|
nexdv |
⊢ ( ∀ 𝑦 𝑦 = 𝑧 → ¬ ∃ 𝑥 ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) ) |
7 |
6
|
pm2.21d |
⊢ ( ∀ 𝑦 𝑦 = 𝑧 → ( ∃ 𝑥 ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) ) |
8 |
7
|
axc4i |
⊢ ( ∀ 𝑦 𝑦 = 𝑧 → ∀ 𝑦 ( ∃ 𝑥 ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) ) |
9 |
8
|
19.8ad |
⊢ ( ∀ 𝑦 𝑦 = 𝑧 → ∃ 𝑥 ∀ 𝑦 ( ∃ 𝑥 ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) ) |
10 |
|
zfun |
⊢ ∃ 𝑥 ∀ 𝑤 ( ∃ 𝑥 ( 𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) → 𝑤 ∈ 𝑥 ) |
11 |
|
nfnae |
⊢ Ⅎ 𝑦 ¬ ∀ 𝑦 𝑦 = 𝑧 |
12 |
|
nfnae |
⊢ Ⅎ 𝑥 ¬ ∀ 𝑦 𝑦 = 𝑧 |
13 |
|
nfvd |
⊢ ( ¬ ∀ 𝑦 𝑦 = 𝑧 → Ⅎ 𝑦 𝑤 ∈ 𝑥 ) |
14 |
|
nfcvf |
⊢ ( ¬ ∀ 𝑦 𝑦 = 𝑧 → Ⅎ 𝑦 𝑧 ) |
15 |
14
|
nfcrd |
⊢ ( ¬ ∀ 𝑦 𝑦 = 𝑧 → Ⅎ 𝑦 𝑥 ∈ 𝑧 ) |
16 |
13 15
|
nfand |
⊢ ( ¬ ∀ 𝑦 𝑦 = 𝑧 → Ⅎ 𝑦 ( 𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) ) |
17 |
12 16
|
nfexd |
⊢ ( ¬ ∀ 𝑦 𝑦 = 𝑧 → Ⅎ 𝑦 ∃ 𝑥 ( 𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) ) |
18 |
17 13
|
nfimd |
⊢ ( ¬ ∀ 𝑦 𝑦 = 𝑧 → Ⅎ 𝑦 ( ∃ 𝑥 ( 𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) → 𝑤 ∈ 𝑥 ) ) |
19 |
|
elequ1 |
⊢ ( 𝑤 = 𝑦 → ( 𝑤 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥 ) ) |
20 |
19
|
anbi1d |
⊢ ( 𝑤 = 𝑦 → ( ( 𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) ↔ ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) ) ) |
21 |
20
|
exbidv |
⊢ ( 𝑤 = 𝑦 → ( ∃ 𝑥 ( 𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) ↔ ∃ 𝑥 ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) ) ) |
22 |
21 19
|
imbi12d |
⊢ ( 𝑤 = 𝑦 → ( ( ∃ 𝑥 ( 𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) → 𝑤 ∈ 𝑥 ) ↔ ( ∃ 𝑥 ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) ) ) |
23 |
22
|
a1i |
⊢ ( ¬ ∀ 𝑦 𝑦 = 𝑧 → ( 𝑤 = 𝑦 → ( ( ∃ 𝑥 ( 𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) → 𝑤 ∈ 𝑥 ) ↔ ( ∃ 𝑥 ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) ) ) ) |
24 |
11 18 23
|
cbvald |
⊢ ( ¬ ∀ 𝑦 𝑦 = 𝑧 → ( ∀ 𝑤 ( ∃ 𝑥 ( 𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) → 𝑤 ∈ 𝑥 ) ↔ ∀ 𝑦 ( ∃ 𝑥 ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) ) ) |
25 |
24
|
exbidv |
⊢ ( ¬ ∀ 𝑦 𝑦 = 𝑧 → ( ∃ 𝑥 ∀ 𝑤 ( ∃ 𝑥 ( 𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) → 𝑤 ∈ 𝑥 ) ↔ ∃ 𝑥 ∀ 𝑦 ( ∃ 𝑥 ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) ) ) |
26 |
10 25
|
mpbii |
⊢ ( ¬ ∀ 𝑦 𝑦 = 𝑧 → ∃ 𝑥 ∀ 𝑦 ( ∃ 𝑥 ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) ) |
27 |
9 26
|
pm2.61i |
⊢ ∃ 𝑥 ∀ 𝑦 ( ∃ 𝑥 ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) |