Step |
Hyp |
Ref |
Expression |
1 |
|
rmoi2.1 |
⊢ ( 𝑥 = 𝐵 → ( 𝜓 ↔ 𝜒 ) ) |
2 |
|
rmoi2.2 |
⊢ ( 𝜑 → 𝐵 ∈ 𝐴 ) |
3 |
|
rmoi2.3 |
⊢ ( 𝜑 → ∃* 𝑥 ∈ 𝐴 𝜓 ) |
4 |
|
rmoi2.4 |
⊢ ( 𝜑 → 𝑥 ∈ 𝐴 ) |
5 |
|
rmoi2.5 |
⊢ ( 𝜑 → 𝜓 ) |
6 |
|
df-rmo |
⊢ ( ∃* 𝑥 ∈ 𝐴 𝜓 ↔ ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ) |
7 |
3 6
|
sylib |
⊢ ( 𝜑 → ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ) |
8 |
|
eleq1 |
⊢ ( 𝑥 = 𝐵 → ( 𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴 ) ) |
9 |
8 1
|
anbi12d |
⊢ ( 𝑥 = 𝐵 → ( ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ↔ ( 𝐵 ∈ 𝐴 ∧ 𝜒 ) ) ) |
10 |
9
|
mob2 |
⊢ ( ( 𝐵 ∈ 𝐴 ∧ ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ) → ( 𝑥 = 𝐵 ↔ ( 𝐵 ∈ 𝐴 ∧ 𝜒 ) ) ) |
11 |
2 7 4 5 10
|
syl112anc |
⊢ ( 𝜑 → ( 𝑥 = 𝐵 ↔ ( 𝐵 ∈ 𝐴 ∧ 𝜒 ) ) ) |
12 |
2 11
|
mpbirand |
⊢ ( 𝜑 → ( 𝑥 = 𝐵 ↔ 𝜒 ) ) |