Step |
Hyp |
Ref |
Expression |
1 |
|
inintabd.x |
⊢ ( 𝜑 → ∃ 𝑥 𝜓 ) |
2 |
|
pm5.5 |
⊢ ( ∃ 𝑥 𝜓 → ( ( ∃ 𝑥 𝜓 → 𝑢 ∈ 𝐴 ) ↔ 𝑢 ∈ 𝐴 ) ) |
3 |
1 2
|
syl |
⊢ ( 𝜑 → ( ( ∃ 𝑥 𝜓 → 𝑢 ∈ 𝐴 ) ↔ 𝑢 ∈ 𝐴 ) ) |
4 |
3
|
bicomd |
⊢ ( 𝜑 → ( 𝑢 ∈ 𝐴 ↔ ( ∃ 𝑥 𝜓 → 𝑢 ∈ 𝐴 ) ) ) |
5 |
4
|
anbi1d |
⊢ ( 𝜑 → ( ( 𝑢 ∈ 𝐴 ∧ ∀ 𝑥 ( 𝜓 → 𝑢 ∈ 𝑥 ) ) ↔ ( ( ∃ 𝑥 𝜓 → 𝑢 ∈ 𝐴 ) ∧ ∀ 𝑥 ( 𝜓 → 𝑢 ∈ 𝑥 ) ) ) ) |
6 |
|
elinintab |
⊢ ( 𝑢 ∈ ( 𝐴 ∩ ∩ { 𝑥 ∣ 𝜓 } ) ↔ ( 𝑢 ∈ 𝐴 ∧ ∀ 𝑥 ( 𝜓 → 𝑢 ∈ 𝑥 ) ) ) |
7 |
|
elinintrab |
⊢ ( 𝑢 ∈ V → ( 𝑢 ∈ ∩ { 𝑤 ∈ 𝒫 𝐴 ∣ ∃ 𝑥 ( 𝑤 = ( 𝐴 ∩ 𝑥 ) ∧ 𝜓 ) } ↔ ( ( ∃ 𝑥 𝜓 → 𝑢 ∈ 𝐴 ) ∧ ∀ 𝑥 ( 𝜓 → 𝑢 ∈ 𝑥 ) ) ) ) |
8 |
7
|
elv |
⊢ ( 𝑢 ∈ ∩ { 𝑤 ∈ 𝒫 𝐴 ∣ ∃ 𝑥 ( 𝑤 = ( 𝐴 ∩ 𝑥 ) ∧ 𝜓 ) } ↔ ( ( ∃ 𝑥 𝜓 → 𝑢 ∈ 𝐴 ) ∧ ∀ 𝑥 ( 𝜓 → 𝑢 ∈ 𝑥 ) ) ) |
9 |
5 6 8
|
3bitr4g |
⊢ ( 𝜑 → ( 𝑢 ∈ ( 𝐴 ∩ ∩ { 𝑥 ∣ 𝜓 } ) ↔ 𝑢 ∈ ∩ { 𝑤 ∈ 𝒫 𝐴 ∣ ∃ 𝑥 ( 𝑤 = ( 𝐴 ∩ 𝑥 ) ∧ 𝜓 ) } ) ) |
10 |
9
|
eqrdv |
⊢ ( 𝜑 → ( 𝐴 ∩ ∩ { 𝑥 ∣ 𝜓 } ) = ∩ { 𝑤 ∈ 𝒫 𝐴 ∣ ∃ 𝑥 ( 𝑤 = ( 𝐴 ∩ 𝑥 ) ∧ 𝜓 ) } ) |