Step |
Hyp |
Ref |
Expression |
1 |
|
nfnae |
⊢ Ⅎ 𝑥 ¬ ∀ 𝑥 𝑥 = 𝑦 |
2 |
|
nfnae |
⊢ Ⅎ 𝑥 ¬ ∀ 𝑥 𝑥 = 𝑧 |
3 |
1 2
|
nfan |
⊢ Ⅎ 𝑥 ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) |
4 |
|
nfcvf |
⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑥 𝑦 ) |
5 |
4
|
adantr |
⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) → Ⅎ 𝑥 𝑦 ) |
6 |
5
|
nfcrd |
⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) → Ⅎ 𝑥 𝑤 ∈ 𝑦 ) |
7 |
|
nfcvf |
⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑧 → Ⅎ 𝑥 𝑧 ) |
8 |
7
|
adantl |
⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) → Ⅎ 𝑥 𝑧 ) |
9 |
8
|
nfcrd |
⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) → Ⅎ 𝑥 𝑤 ∈ 𝑧 ) |
10 |
6 9
|
nfbid |
⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) → Ⅎ 𝑥 ( 𝑤 ∈ 𝑦 ↔ 𝑤 ∈ 𝑧 ) ) |
11 |
|
elequ1 |
⊢ ( 𝑤 = 𝑥 → ( 𝑤 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦 ) ) |
12 |
|
elequ1 |
⊢ ( 𝑤 = 𝑥 → ( 𝑤 ∈ 𝑧 ↔ 𝑥 ∈ 𝑧 ) ) |
13 |
11 12
|
bibi12d |
⊢ ( 𝑤 = 𝑥 → ( ( 𝑤 ∈ 𝑦 ↔ 𝑤 ∈ 𝑧 ) ↔ ( 𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧 ) ) ) |
14 |
13
|
a1i |
⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) → ( 𝑤 = 𝑥 → ( ( 𝑤 ∈ 𝑦 ↔ 𝑤 ∈ 𝑧 ) ↔ ( 𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧 ) ) ) ) |
15 |
3 10 14
|
cbvald |
⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) → ( ∀ 𝑤 ( 𝑤 ∈ 𝑦 ↔ 𝑤 ∈ 𝑧 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧 ) ) ) |
16 |
|
axextg |
⊢ ( ∀ 𝑤 ( 𝑤 ∈ 𝑦 ↔ 𝑤 ∈ 𝑧 ) → 𝑦 = 𝑧 ) |
17 |
15 16
|
syl6bir |
⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) → ( ∀ 𝑥 ( 𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧 ) → 𝑦 = 𝑧 ) ) |
18 |
|
19.8a |
⊢ ( 𝑦 = 𝑧 → ∃ 𝑥 𝑦 = 𝑧 ) |
19 |
17 18
|
syl6 |
⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) → ( ∀ 𝑥 ( 𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧 ) → ∃ 𝑥 𝑦 = 𝑧 ) ) |
20 |
19
|
ex |
⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( ¬ ∀ 𝑥 𝑥 = 𝑧 → ( ∀ 𝑥 ( 𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧 ) → ∃ 𝑥 𝑦 = 𝑧 ) ) ) |
21 |
|
ax6e |
⊢ ∃ 𝑥 𝑥 = 𝑧 |
22 |
|
ax7 |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 = 𝑧 → 𝑦 = 𝑧 ) ) |
23 |
22
|
aleximi |
⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∃ 𝑥 𝑥 = 𝑧 → ∃ 𝑥 𝑦 = 𝑧 ) ) |
24 |
21 23
|
mpi |
⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ∃ 𝑥 𝑦 = 𝑧 ) |
25 |
24
|
a1d |
⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 ( 𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧 ) → ∃ 𝑥 𝑦 = 𝑧 ) ) |
26 |
|
ax6e |
⊢ ∃ 𝑥 𝑥 = 𝑦 |
27 |
|
ax7 |
⊢ ( 𝑥 = 𝑧 → ( 𝑥 = 𝑦 → 𝑧 = 𝑦 ) ) |
28 |
|
equcomi |
⊢ ( 𝑧 = 𝑦 → 𝑦 = 𝑧 ) |
29 |
27 28
|
syl6 |
⊢ ( 𝑥 = 𝑧 → ( 𝑥 = 𝑦 → 𝑦 = 𝑧 ) ) |
30 |
29
|
aleximi |
⊢ ( ∀ 𝑥 𝑥 = 𝑧 → ( ∃ 𝑥 𝑥 = 𝑦 → ∃ 𝑥 𝑦 = 𝑧 ) ) |
31 |
26 30
|
mpi |
⊢ ( ∀ 𝑥 𝑥 = 𝑧 → ∃ 𝑥 𝑦 = 𝑧 ) |
32 |
31
|
a1d |
⊢ ( ∀ 𝑥 𝑥 = 𝑧 → ( ∀ 𝑥 ( 𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧 ) → ∃ 𝑥 𝑦 = 𝑧 ) ) |
33 |
20 25 32
|
pm2.61ii |
⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧 ) → ∃ 𝑥 𝑦 = 𝑧 ) |
34 |
33
|
19.35ri |
⊢ ∃ 𝑥 ( ( 𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧 ) → 𝑦 = 𝑧 ) |