Step |
Hyp |
Ref |
Expression |
1 |
|
riotasvd.1 |
⊢ ( 𝜑 → 𝐷 = ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝜓 → 𝑥 = 𝐶 ) ) ) |
2 |
|
riotasvd.2 |
⊢ ( 𝜑 → 𝐷 ∈ 𝐴 ) |
3 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑉 ) → 𝐷 = ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝜓 → 𝑥 = 𝐶 ) ) ) |
4 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑉 ) → 𝐷 ∈ 𝐴 ) |
5 |
3 4
|
eqeltrrd |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑉 ) → ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝜓 → 𝑥 = 𝐶 ) ) ∈ 𝐴 ) |
6 |
|
riotaclbgBAD |
⊢ ( 𝐴 ∈ 𝑉 → ( ∃! 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝜓 → 𝑥 = 𝐶 ) ↔ ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝜓 → 𝑥 = 𝐶 ) ) ∈ 𝐴 ) ) |
7 |
6
|
adantl |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑉 ) → ( ∃! 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝜓 → 𝑥 = 𝐶 ) ↔ ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝜓 → 𝑥 = 𝐶 ) ) ∈ 𝐴 ) ) |
8 |
5 7
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑉 ) → ∃! 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝜓 → 𝑥 = 𝐶 ) ) |
9 |
|
riotasbc |
⊢ ( ∃! 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝜓 → 𝑥 = 𝐶 ) → [ ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝜓 → 𝑥 = 𝐶 ) ) / 𝑥 ] ∀ 𝑦 ∈ 𝐵 ( 𝜓 → 𝑥 = 𝐶 ) ) |
10 |
8 9
|
syl |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑉 ) → [ ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝜓 → 𝑥 = 𝐶 ) ) / 𝑥 ] ∀ 𝑦 ∈ 𝐵 ( 𝜓 → 𝑥 = 𝐶 ) ) |
11 |
|
eqeq1 |
⊢ ( 𝑥 = 𝑧 → ( 𝑥 = 𝐶 ↔ 𝑧 = 𝐶 ) ) |
12 |
11
|
imbi2d |
⊢ ( 𝑥 = 𝑧 → ( ( 𝜓 → 𝑥 = 𝐶 ) ↔ ( 𝜓 → 𝑧 = 𝐶 ) ) ) |
13 |
12
|
ralbidv |
⊢ ( 𝑥 = 𝑧 → ( ∀ 𝑦 ∈ 𝐵 ( 𝜓 → 𝑥 = 𝐶 ) ↔ ∀ 𝑦 ∈ 𝐵 ( 𝜓 → 𝑧 = 𝐶 ) ) ) |
14 |
|
nfra1 |
⊢ Ⅎ 𝑦 ∀ 𝑦 ∈ 𝐵 ( 𝜓 → 𝑥 = 𝐶 ) |
15 |
|
nfcv |
⊢ Ⅎ 𝑦 𝐴 |
16 |
14 15
|
nfriota |
⊢ Ⅎ 𝑦 ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝜓 → 𝑥 = 𝐶 ) ) |
17 |
16
|
nfeq2 |
⊢ Ⅎ 𝑦 𝑧 = ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝜓 → 𝑥 = 𝐶 ) ) |
18 |
|
eqeq1 |
⊢ ( 𝑧 = ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝜓 → 𝑥 = 𝐶 ) ) → ( 𝑧 = 𝐶 ↔ ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝜓 → 𝑥 = 𝐶 ) ) = 𝐶 ) ) |
19 |
18
|
imbi2d |
⊢ ( 𝑧 = ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝜓 → 𝑥 = 𝐶 ) ) → ( ( 𝜓 → 𝑧 = 𝐶 ) ↔ ( 𝜓 → ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝜓 → 𝑥 = 𝐶 ) ) = 𝐶 ) ) ) |
20 |
17 19
|
ralbid |
⊢ ( 𝑧 = ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝜓 → 𝑥 = 𝐶 ) ) → ( ∀ 𝑦 ∈ 𝐵 ( 𝜓 → 𝑧 = 𝐶 ) ↔ ∀ 𝑦 ∈ 𝐵 ( 𝜓 → ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝜓 → 𝑥 = 𝐶 ) ) = 𝐶 ) ) ) |
21 |
13 20
|
sbcie2g |
⊢ ( ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝜓 → 𝑥 = 𝐶 ) ) ∈ 𝐴 → ( [ ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝜓 → 𝑥 = 𝐶 ) ) / 𝑥 ] ∀ 𝑦 ∈ 𝐵 ( 𝜓 → 𝑥 = 𝐶 ) ↔ ∀ 𝑦 ∈ 𝐵 ( 𝜓 → ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝜓 → 𝑥 = 𝐶 ) ) = 𝐶 ) ) ) |
22 |
5 21
|
syl |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑉 ) → ( [ ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝜓 → 𝑥 = 𝐶 ) ) / 𝑥 ] ∀ 𝑦 ∈ 𝐵 ( 𝜓 → 𝑥 = 𝐶 ) ↔ ∀ 𝑦 ∈ 𝐵 ( 𝜓 → ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝜓 → 𝑥 = 𝐶 ) ) = 𝐶 ) ) ) |
23 |
10 22
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑉 ) → ∀ 𝑦 ∈ 𝐵 ( 𝜓 → ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝜓 → 𝑥 = 𝐶 ) ) = 𝐶 ) ) |
24 |
|
rsp |
⊢ ( ∀ 𝑦 ∈ 𝐵 ( 𝜓 → ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝜓 → 𝑥 = 𝐶 ) ) = 𝐶 ) → ( 𝑦 ∈ 𝐵 → ( 𝜓 → ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝜓 → 𝑥 = 𝐶 ) ) = 𝐶 ) ) ) |
25 |
23 24
|
syl |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑉 ) → ( 𝑦 ∈ 𝐵 → ( 𝜓 → ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝜓 → 𝑥 = 𝐶 ) ) = 𝐶 ) ) ) |
26 |
25
|
impd |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑉 ) → ( ( 𝑦 ∈ 𝐵 ∧ 𝜓 ) → ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝜓 → 𝑥 = 𝐶 ) ) = 𝐶 ) ) |
27 |
3
|
eqeq1d |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑉 ) → ( 𝐷 = 𝐶 ↔ ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝜓 → 𝑥 = 𝐶 ) ) = 𝐶 ) ) |
28 |
26 27
|
sylibrd |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑉 ) → ( ( 𝑦 ∈ 𝐵 ∧ 𝜓 ) → 𝐷 = 𝐶 ) ) |