Step |
Hyp |
Ref |
Expression |
1 |
|
nfv |
⊢ Ⅎ 𝑧 ∃ 𝑥 ∈ 𝐴 𝜑 |
2 |
|
nfcv |
⊢ Ⅎ 𝑦 𝐴 |
3 |
|
nfs1v |
⊢ Ⅎ 𝑦 [ 𝑧 / 𝑦 ] 𝜑 |
4 |
2 3
|
nfrex |
⊢ Ⅎ 𝑦 ∃ 𝑥 ∈ 𝐴 [ 𝑧 / 𝑦 ] 𝜑 |
5 |
|
sbequ12 |
⊢ ( 𝑦 = 𝑧 → ( 𝜑 ↔ [ 𝑧 / 𝑦 ] 𝜑 ) ) |
6 |
5
|
rexbidv |
⊢ ( 𝑦 = 𝑧 → ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑥 ∈ 𝐴 [ 𝑧 / 𝑦 ] 𝜑 ) ) |
7 |
1 4 6
|
cbvabw |
⊢ { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝜑 } = { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 [ 𝑧 / 𝑦 ] 𝜑 } |
8 |
|
df-clab |
⊢ ( 𝑧 ∈ { 𝑦 ∣ 𝜑 } ↔ [ 𝑧 / 𝑦 ] 𝜑 ) |
9 |
8
|
rexbii |
⊢ ( ∃ 𝑥 ∈ 𝐴 𝑧 ∈ { 𝑦 ∣ 𝜑 } ↔ ∃ 𝑥 ∈ 𝐴 [ 𝑧 / 𝑦 ] 𝜑 ) |
10 |
9
|
abbii |
⊢ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 ∈ { 𝑦 ∣ 𝜑 } } = { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 [ 𝑧 / 𝑦 ] 𝜑 } |
11 |
7 10
|
eqtr4i |
⊢ { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝜑 } = { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 ∈ { 𝑦 ∣ 𝜑 } } |
12 |
|
df-iun |
⊢ ∪ 𝑥 ∈ 𝐴 { 𝑦 ∣ 𝜑 } = { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 ∈ { 𝑦 ∣ 𝜑 } } |
13 |
|
iunexg |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ∀ 𝑥 ∈ 𝐴 { 𝑦 ∣ 𝜑 } ∈ 𝑊 ) → ∪ 𝑥 ∈ 𝐴 { 𝑦 ∣ 𝜑 } ∈ V ) |
14 |
12 13
|
eqeltrrid |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ∀ 𝑥 ∈ 𝐴 { 𝑦 ∣ 𝜑 } ∈ 𝑊 ) → { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 ∈ { 𝑦 ∣ 𝜑 } } ∈ V ) |
15 |
11 14
|
eqeltrid |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ∀ 𝑥 ∈ 𝐴 { 𝑦 ∣ 𝜑 } ∈ 𝑊 ) → { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝜑 } ∈ V ) |