Step |
Hyp |
Ref |
Expression |
1 |
|
velsn |
⊢ ( 𝑥 ∈ { 𝑁 } ↔ 𝑥 = 𝑁 ) |
2 |
|
eleq1a |
⊢ ( 𝑁 ∈ 𝑉 → ( 𝑥 = 𝑁 → 𝑥 ∈ 𝑉 ) ) |
3 |
2
|
pm4.71rd |
⊢ ( 𝑁 ∈ 𝑉 → ( 𝑥 = 𝑁 ↔ ( 𝑥 ∈ 𝑉 ∧ 𝑥 = 𝑁 ) ) ) |
4 |
1 3
|
bitrid |
⊢ ( 𝑁 ∈ 𝑉 → ( 𝑥 ∈ { 𝑁 } ↔ ( 𝑥 ∈ 𝑉 ∧ 𝑥 = 𝑁 ) ) ) |
5 |
4
|
anbi1d |
⊢ ( 𝑁 ∈ 𝑉 → ( ( 𝑥 ∈ { 𝑁 } ∧ 𝜓 ) ↔ ( ( 𝑥 ∈ 𝑉 ∧ 𝑥 = 𝑁 ) ∧ 𝜓 ) ) ) |
6 |
|
anass |
⊢ ( ( ( 𝑥 ∈ 𝑉 ∧ 𝑥 = 𝑁 ) ∧ 𝜓 ) ↔ ( 𝑥 ∈ 𝑉 ∧ ( 𝑥 = 𝑁 ∧ 𝜓 ) ) ) |
7 |
5 6
|
bitrdi |
⊢ ( 𝑁 ∈ 𝑉 → ( ( 𝑥 ∈ { 𝑁 } ∧ 𝜓 ) ↔ ( 𝑥 ∈ 𝑉 ∧ ( 𝑥 = 𝑁 ∧ 𝜓 ) ) ) ) |
8 |
7
|
abbidv |
⊢ ( 𝑁 ∈ 𝑉 → { 𝑥 ∣ ( 𝑥 ∈ { 𝑁 } ∧ 𝜓 ) } = { 𝑥 ∣ ( 𝑥 ∈ 𝑉 ∧ ( 𝑥 = 𝑁 ∧ 𝜓 ) ) } ) |
9 |
|
df-rab |
⊢ { 𝑥 ∈ { 𝑁 } ∣ 𝜓 } = { 𝑥 ∣ ( 𝑥 ∈ { 𝑁 } ∧ 𝜓 ) } |
10 |
|
df-rab |
⊢ { 𝑥 ∈ 𝑉 ∣ ( 𝑥 = 𝑁 ∧ 𝜓 ) } = { 𝑥 ∣ ( 𝑥 ∈ 𝑉 ∧ ( 𝑥 = 𝑁 ∧ 𝜓 ) ) } |
11 |
8 9 10
|
3eqtr4g |
⊢ ( 𝑁 ∈ 𝑉 → { 𝑥 ∈ { 𝑁 } ∣ 𝜓 } = { 𝑥 ∈ 𝑉 ∣ ( 𝑥 = 𝑁 ∧ 𝜓 ) } ) |