Step |
Hyp |
Ref |
Expression |
1 |
|
rmounid.1 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ¬ 𝜓 ) |
2 |
1
|
ex |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐵 → ¬ 𝜓 ) ) |
3 |
2
|
con2d |
⊢ ( 𝜑 → ( 𝜓 → ¬ 𝑥 ∈ 𝐵 ) ) |
4 |
3
|
imp |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ¬ 𝑥 ∈ 𝐵 ) |
5 |
|
biorf |
⊢ ( ¬ 𝑥 ∈ 𝐵 → ( 𝑥 ∈ 𝐴 ↔ ( 𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐴 ) ) ) |
6 |
|
orcom |
⊢ ( ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 ) ↔ ( 𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐴 ) ) |
7 |
5 6
|
bitr4di |
⊢ ( ¬ 𝑥 ∈ 𝐵 → ( 𝑥 ∈ 𝐴 ↔ ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 ) ) ) |
8 |
4 7
|
syl |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝑥 ∈ 𝐴 ↔ ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 ) ) ) |
9 |
|
elun |
⊢ ( 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) ↔ ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 ) ) |
10 |
8 9
|
bitr4di |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) ) ) |
11 |
10
|
pm5.32da |
⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝑥 ∈ 𝐴 ) ↔ ( 𝜓 ∧ 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) ) ) ) |
12 |
11
|
biancomd |
⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝑥 ∈ 𝐴 ) ↔ ( 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) ∧ 𝜓 ) ) ) |
13 |
12
|
bicomd |
⊢ ( 𝜑 → ( ( 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) ∧ 𝜓 ) ↔ ( 𝜓 ∧ 𝑥 ∈ 𝐴 ) ) ) |
14 |
13
|
biancomd |
⊢ ( 𝜑 → ( ( 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) ∧ 𝜓 ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ) ) |
15 |
14
|
mobidv |
⊢ ( 𝜑 → ( ∃* 𝑥 ( 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) ∧ 𝜓 ) ↔ ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ) ) |
16 |
|
df-rmo |
⊢ ( ∃* 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) 𝜓 ↔ ∃* 𝑥 ( 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) ∧ 𝜓 ) ) |
17 |
|
df-rmo |
⊢ ( ∃* 𝑥 ∈ 𝐴 𝜓 ↔ ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ) |
18 |
15 16 17
|
3bitr4g |
⊢ ( 𝜑 → ( ∃* 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) 𝜓 ↔ ∃* 𝑥 ∈ 𝐴 𝜓 ) ) |