Step |
Hyp |
Ref |
Expression |
1 |
|
eleq2 |
⊢ ( ( 𝐴 ∩ 𝐶 ) = ( 𝐵 ∩ 𝐶 ) → ( 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ↔ 𝑥 ∈ ( 𝐵 ∩ 𝐶 ) ) ) |
2 |
|
elin |
⊢ ( 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶 ) ) |
3 |
|
elin |
⊢ ( 𝑥 ∈ ( 𝐵 ∩ 𝐶 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶 ) ) |
4 |
1 2 3
|
3bitr3g |
⊢ ( ( 𝐴 ∩ 𝐶 ) = ( 𝐵 ∩ 𝐶 ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶 ) ) ) |
5 |
|
iba |
⊢ ( 𝑥 ∈ 𝐶 → ( 𝑥 ∈ 𝐴 ↔ ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶 ) ) ) |
6 |
|
iba |
⊢ ( 𝑥 ∈ 𝐶 → ( 𝑥 ∈ 𝐵 ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶 ) ) ) |
7 |
5 6
|
bibi12d |
⊢ ( 𝑥 ∈ 𝐶 → ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ↔ ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶 ) ) ) ) |
8 |
4 7
|
syl5ibr |
⊢ ( 𝑥 ∈ 𝐶 → ( ( 𝐴 ∩ 𝐶 ) = ( 𝐵 ∩ 𝐶 ) → ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ) ) |
9 |
8
|
adantld |
⊢ ( 𝑥 ∈ 𝐶 → ( ( ( 𝐴 ∪ 𝐶 ) = ( 𝐵 ∪ 𝐶 ) ∧ ( 𝐴 ∩ 𝐶 ) = ( 𝐵 ∩ 𝐶 ) ) → ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ) ) |
10 |
|
uncom |
⊢ ( 𝐴 ∪ 𝐶 ) = ( 𝐶 ∪ 𝐴 ) |
11 |
|
uncom |
⊢ ( 𝐵 ∪ 𝐶 ) = ( 𝐶 ∪ 𝐵 ) |
12 |
10 11
|
eqeq12i |
⊢ ( ( 𝐴 ∪ 𝐶 ) = ( 𝐵 ∪ 𝐶 ) ↔ ( 𝐶 ∪ 𝐴 ) = ( 𝐶 ∪ 𝐵 ) ) |
13 |
|
eleq2 |
⊢ ( ( 𝐶 ∪ 𝐴 ) = ( 𝐶 ∪ 𝐵 ) → ( 𝑥 ∈ ( 𝐶 ∪ 𝐴 ) ↔ 𝑥 ∈ ( 𝐶 ∪ 𝐵 ) ) ) |
14 |
12 13
|
sylbi |
⊢ ( ( 𝐴 ∪ 𝐶 ) = ( 𝐵 ∪ 𝐶 ) → ( 𝑥 ∈ ( 𝐶 ∪ 𝐴 ) ↔ 𝑥 ∈ ( 𝐶 ∪ 𝐵 ) ) ) |
15 |
|
elun |
⊢ ( 𝑥 ∈ ( 𝐶 ∪ 𝐴 ) ↔ ( 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴 ) ) |
16 |
|
elun |
⊢ ( 𝑥 ∈ ( 𝐶 ∪ 𝐵 ) ↔ ( 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐵 ) ) |
17 |
14 15 16
|
3bitr3g |
⊢ ( ( 𝐴 ∪ 𝐶 ) = ( 𝐵 ∪ 𝐶 ) → ( ( 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴 ) ↔ ( 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐵 ) ) ) |
18 |
|
biorf |
⊢ ( ¬ 𝑥 ∈ 𝐶 → ( 𝑥 ∈ 𝐴 ↔ ( 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴 ) ) ) |
19 |
|
biorf |
⊢ ( ¬ 𝑥 ∈ 𝐶 → ( 𝑥 ∈ 𝐵 ↔ ( 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐵 ) ) ) |
20 |
18 19
|
bibi12d |
⊢ ( ¬ 𝑥 ∈ 𝐶 → ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ↔ ( ( 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴 ) ↔ ( 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐵 ) ) ) ) |
21 |
17 20
|
syl5ibr |
⊢ ( ¬ 𝑥 ∈ 𝐶 → ( ( 𝐴 ∪ 𝐶 ) = ( 𝐵 ∪ 𝐶 ) → ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ) ) |
22 |
21
|
adantrd |
⊢ ( ¬ 𝑥 ∈ 𝐶 → ( ( ( 𝐴 ∪ 𝐶 ) = ( 𝐵 ∪ 𝐶 ) ∧ ( 𝐴 ∩ 𝐶 ) = ( 𝐵 ∩ 𝐶 ) ) → ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ) ) |
23 |
9 22
|
pm2.61i |
⊢ ( ( ( 𝐴 ∪ 𝐶 ) = ( 𝐵 ∪ 𝐶 ) ∧ ( 𝐴 ∩ 𝐶 ) = ( 𝐵 ∩ 𝐶 ) ) → ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ) |
24 |
23
|
eqrdv |
⊢ ( ( ( 𝐴 ∪ 𝐶 ) = ( 𝐵 ∪ 𝐶 ) ∧ ( 𝐴 ∩ 𝐶 ) = ( 𝐵 ∩ 𝐶 ) ) → 𝐴 = 𝐵 ) |
25 |
|
uneq1 |
⊢ ( 𝐴 = 𝐵 → ( 𝐴 ∪ 𝐶 ) = ( 𝐵 ∪ 𝐶 ) ) |
26 |
|
ineq1 |
⊢ ( 𝐴 = 𝐵 → ( 𝐴 ∩ 𝐶 ) = ( 𝐵 ∩ 𝐶 ) ) |
27 |
25 26
|
jca |
⊢ ( 𝐴 = 𝐵 → ( ( 𝐴 ∪ 𝐶 ) = ( 𝐵 ∪ 𝐶 ) ∧ ( 𝐴 ∩ 𝐶 ) = ( 𝐵 ∩ 𝐶 ) ) ) |
28 |
24 27
|
impbii |
⊢ ( ( ( 𝐴 ∪ 𝐶 ) = ( 𝐵 ∪ 𝐶 ) ∧ ( 𝐴 ∩ 𝐶 ) = ( 𝐵 ∩ 𝐶 ) ) ↔ 𝐴 = 𝐵 ) |