Step |
Hyp |
Ref |
Expression |
1 |
|
dfcleq |
⊢ ( 𝐴 = 𝐵 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ) |
2 |
1
|
biimpi |
⊢ ( 𝐴 = 𝐵 → ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ) |
3 |
|
anbi1 |
⊢ ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) ) ) |
4 |
3
|
imbi1d |
⊢ ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) → ( ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝑥 = 𝑧 ) ↔ ( ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) → 𝑥 = 𝑧 ) ) ) |
5 |
4
|
alimi |
⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) → ∀ 𝑥 ( ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝑥 = 𝑧 ) ↔ ( ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) → 𝑥 = 𝑧 ) ) ) |
6 |
|
albi |
⊢ ( ∀ 𝑥 ( ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝑥 = 𝑧 ) ↔ ( ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) → 𝑥 = 𝑧 ) ) → ( ∀ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝑥 = 𝑧 ) ↔ ∀ 𝑥 ( ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) → 𝑥 = 𝑧 ) ) ) |
7 |
2 5 6
|
3syl |
⊢ ( 𝐴 = 𝐵 → ( ∀ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝑥 = 𝑧 ) ↔ ∀ 𝑥 ( ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) → 𝑥 = 𝑧 ) ) ) |
8 |
7
|
exbidv |
⊢ ( 𝐴 = 𝐵 → ( ∃ 𝑧 ∀ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝑥 = 𝑧 ) ↔ ∃ 𝑧 ∀ 𝑥 ( ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) → 𝑥 = 𝑧 ) ) ) |
9 |
|
df-mo |
⊢ ( ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ∃ 𝑧 ∀ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝑥 = 𝑧 ) ) |
10 |
|
df-mo |
⊢ ( ∃* 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) ↔ ∃ 𝑧 ∀ 𝑥 ( ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) → 𝑥 = 𝑧 ) ) |
11 |
8 9 10
|
3bitr4g |
⊢ ( 𝐴 = 𝐵 → ( ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ∃* 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) ) ) |
12 |
|
df-rmo |
⊢ ( ∃* 𝑥 ∈ 𝐴 𝜑 ↔ ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) |
13 |
|
df-rmo |
⊢ ( ∃* 𝑥 ∈ 𝐵 𝜑 ↔ ∃* 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) ) |
14 |
11 12 13
|
3bitr4g |
⊢ ( 𝐴 = 𝐵 → ( ∃* 𝑥 ∈ 𝐴 𝜑 ↔ ∃* 𝑥 ∈ 𝐵 𝜑 ) ) |