Step |
Hyp |
Ref |
Expression |
1 |
|
iota5f.1 |
⊢ Ⅎ 𝑥 𝜑 |
2 |
|
iota5f.2 |
⊢ Ⅎ 𝑥 𝐴 |
3 |
|
iota5f.3 |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑉 ) → ( 𝜓 ↔ 𝑥 = 𝐴 ) ) |
4 |
2
|
nfel1 |
⊢ Ⅎ 𝑥 𝐴 ∈ 𝑉 |
5 |
1 4
|
nfan |
⊢ Ⅎ 𝑥 ( 𝜑 ∧ 𝐴 ∈ 𝑉 ) |
6 |
5 3
|
alrimi |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑉 ) → ∀ 𝑥 ( 𝜓 ↔ 𝑥 = 𝐴 ) ) |
7 |
2
|
nfeq2 |
⊢ Ⅎ 𝑥 𝑦 = 𝐴 |
8 |
|
eqeq2 |
⊢ ( 𝑦 = 𝐴 → ( 𝑥 = 𝑦 ↔ 𝑥 = 𝐴 ) ) |
9 |
8
|
bibi2d |
⊢ ( 𝑦 = 𝐴 → ( ( 𝜓 ↔ 𝑥 = 𝑦 ) ↔ ( 𝜓 ↔ 𝑥 = 𝐴 ) ) ) |
10 |
7 9
|
albid |
⊢ ( 𝑦 = 𝐴 → ( ∀ 𝑥 ( 𝜓 ↔ 𝑥 = 𝑦 ) ↔ ∀ 𝑥 ( 𝜓 ↔ 𝑥 = 𝐴 ) ) ) |
11 |
|
eqeq2 |
⊢ ( 𝑦 = 𝐴 → ( ( ℩ 𝑥 𝜓 ) = 𝑦 ↔ ( ℩ 𝑥 𝜓 ) = 𝐴 ) ) |
12 |
10 11
|
imbi12d |
⊢ ( 𝑦 = 𝐴 → ( ( ∀ 𝑥 ( 𝜓 ↔ 𝑥 = 𝑦 ) → ( ℩ 𝑥 𝜓 ) = 𝑦 ) ↔ ( ∀ 𝑥 ( 𝜓 ↔ 𝑥 = 𝐴 ) → ( ℩ 𝑥 𝜓 ) = 𝐴 ) ) ) |
13 |
|
iotaval |
⊢ ( ∀ 𝑥 ( 𝜓 ↔ 𝑥 = 𝑦 ) → ( ℩ 𝑥 𝜓 ) = 𝑦 ) |
14 |
12 13
|
vtoclg |
⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ( 𝜓 ↔ 𝑥 = 𝐴 ) → ( ℩ 𝑥 𝜓 ) = 𝐴 ) ) |
15 |
14
|
adantl |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑉 ) → ( ∀ 𝑥 ( 𝜓 ↔ 𝑥 = 𝐴 ) → ( ℩ 𝑥 𝜓 ) = 𝐴 ) ) |
16 |
6 15
|
mpd |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑉 ) → ( ℩ 𝑥 𝜓 ) = 𝐴 ) |