Step |
Hyp |
Ref |
Expression |
1 |
|
ancom |
⊢ ( ( 𝑧 ∈ 𝑥 ∧ ∃ 𝑦 ∈ 𝐴 𝑥 = ( 𝑦 ∩ 𝐵 ) ) ↔ ( ∃ 𝑦 ∈ 𝐴 𝑥 = ( 𝑦 ∩ 𝐵 ) ∧ 𝑧 ∈ 𝑥 ) ) |
2 |
|
r19.41v |
⊢ ( ∃ 𝑦 ∈ 𝐴 ( 𝑥 = ( 𝑦 ∩ 𝐵 ) ∧ 𝑧 ∈ 𝑥 ) ↔ ( ∃ 𝑦 ∈ 𝐴 𝑥 = ( 𝑦 ∩ 𝐵 ) ∧ 𝑧 ∈ 𝑥 ) ) |
3 |
1 2
|
bitr4i |
⊢ ( ( 𝑧 ∈ 𝑥 ∧ ∃ 𝑦 ∈ 𝐴 𝑥 = ( 𝑦 ∩ 𝐵 ) ) ↔ ∃ 𝑦 ∈ 𝐴 ( 𝑥 = ( 𝑦 ∩ 𝐵 ) ∧ 𝑧 ∈ 𝑥 ) ) |
4 |
3
|
exbii |
⊢ ( ∃ 𝑥 ( 𝑧 ∈ 𝑥 ∧ ∃ 𝑦 ∈ 𝐴 𝑥 = ( 𝑦 ∩ 𝐵 ) ) ↔ ∃ 𝑥 ∃ 𝑦 ∈ 𝐴 ( 𝑥 = ( 𝑦 ∩ 𝐵 ) ∧ 𝑧 ∈ 𝑥 ) ) |
5 |
|
eluniab |
⊢ ( 𝑧 ∈ ∪ { 𝑥 ∣ ∃ 𝑦 ∈ 𝐴 𝑥 = ( 𝑦 ∩ 𝐵 ) } ↔ ∃ 𝑥 ( 𝑧 ∈ 𝑥 ∧ ∃ 𝑦 ∈ 𝐴 𝑥 = ( 𝑦 ∩ 𝐵 ) ) ) |
6 |
|
eluni2 |
⊢ ( 𝑧 ∈ ∪ 𝐴 ↔ ∃ 𝑦 ∈ 𝐴 𝑧 ∈ 𝑦 ) |
7 |
6
|
anbi1i |
⊢ ( ( 𝑧 ∈ ∪ 𝐴 ∧ 𝑧 ∈ 𝐵 ) ↔ ( ∃ 𝑦 ∈ 𝐴 𝑧 ∈ 𝑦 ∧ 𝑧 ∈ 𝐵 ) ) |
8 |
|
elin |
⊢ ( 𝑧 ∈ ( ∪ 𝐴 ∩ 𝐵 ) ↔ ( 𝑧 ∈ ∪ 𝐴 ∧ 𝑧 ∈ 𝐵 ) ) |
9 |
|
r19.41v |
⊢ ( ∃ 𝑦 ∈ 𝐴 ( 𝑧 ∈ 𝑦 ∧ 𝑧 ∈ 𝐵 ) ↔ ( ∃ 𝑦 ∈ 𝐴 𝑧 ∈ 𝑦 ∧ 𝑧 ∈ 𝐵 ) ) |
10 |
7 8 9
|
3bitr4i |
⊢ ( 𝑧 ∈ ( ∪ 𝐴 ∩ 𝐵 ) ↔ ∃ 𝑦 ∈ 𝐴 ( 𝑧 ∈ 𝑦 ∧ 𝑧 ∈ 𝐵 ) ) |
11 |
|
vex |
⊢ 𝑦 ∈ V |
12 |
11
|
inex1 |
⊢ ( 𝑦 ∩ 𝐵 ) ∈ V |
13 |
|
eleq2 |
⊢ ( 𝑥 = ( 𝑦 ∩ 𝐵 ) → ( 𝑧 ∈ 𝑥 ↔ 𝑧 ∈ ( 𝑦 ∩ 𝐵 ) ) ) |
14 |
12 13
|
ceqsexv |
⊢ ( ∃ 𝑥 ( 𝑥 = ( 𝑦 ∩ 𝐵 ) ∧ 𝑧 ∈ 𝑥 ) ↔ 𝑧 ∈ ( 𝑦 ∩ 𝐵 ) ) |
15 |
|
elin |
⊢ ( 𝑧 ∈ ( 𝑦 ∩ 𝐵 ) ↔ ( 𝑧 ∈ 𝑦 ∧ 𝑧 ∈ 𝐵 ) ) |
16 |
14 15
|
bitri |
⊢ ( ∃ 𝑥 ( 𝑥 = ( 𝑦 ∩ 𝐵 ) ∧ 𝑧 ∈ 𝑥 ) ↔ ( 𝑧 ∈ 𝑦 ∧ 𝑧 ∈ 𝐵 ) ) |
17 |
16
|
rexbii |
⊢ ( ∃ 𝑦 ∈ 𝐴 ∃ 𝑥 ( 𝑥 = ( 𝑦 ∩ 𝐵 ) ∧ 𝑧 ∈ 𝑥 ) ↔ ∃ 𝑦 ∈ 𝐴 ( 𝑧 ∈ 𝑦 ∧ 𝑧 ∈ 𝐵 ) ) |
18 |
|
rexcom4 |
⊢ ( ∃ 𝑦 ∈ 𝐴 ∃ 𝑥 ( 𝑥 = ( 𝑦 ∩ 𝐵 ) ∧ 𝑧 ∈ 𝑥 ) ↔ ∃ 𝑥 ∃ 𝑦 ∈ 𝐴 ( 𝑥 = ( 𝑦 ∩ 𝐵 ) ∧ 𝑧 ∈ 𝑥 ) ) |
19 |
10 17 18
|
3bitr2i |
⊢ ( 𝑧 ∈ ( ∪ 𝐴 ∩ 𝐵 ) ↔ ∃ 𝑥 ∃ 𝑦 ∈ 𝐴 ( 𝑥 = ( 𝑦 ∩ 𝐵 ) ∧ 𝑧 ∈ 𝑥 ) ) |
20 |
4 5 19
|
3bitr4ri |
⊢ ( 𝑧 ∈ ( ∪ 𝐴 ∩ 𝐵 ) ↔ 𝑧 ∈ ∪ { 𝑥 ∣ ∃ 𝑦 ∈ 𝐴 𝑥 = ( 𝑦 ∩ 𝐵 ) } ) |
21 |
20
|
eqriv |
⊢ ( ∪ 𝐴 ∩ 𝐵 ) = ∪ { 𝑥 ∣ ∃ 𝑦 ∈ 𝐴 𝑥 = ( 𝑦 ∩ 𝐵 ) } |