Step |
Hyp |
Ref |
Expression |
1 |
|
axrep4v |
⊢ ( ∀ 𝑤 ∃ 𝑦 ∀ 𝑧 ( 𝜑 → 𝑧 = 𝑦 ) → ∃ 𝑦 ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ ∃ 𝑤 ( 𝑤 ∈ 𝑥 ∧ 𝜑 ) ) ) |
2 |
|
df-mo |
⊢ ( ∃* 𝑧 𝜑 ↔ ∃ 𝑦 ∀ 𝑧 ( 𝜑 → 𝑧 = 𝑦 ) ) |
3 |
2
|
albii |
⊢ ( ∀ 𝑤 ∃* 𝑧 𝜑 ↔ ∀ 𝑤 ∃ 𝑦 ∀ 𝑧 ( 𝜑 → 𝑧 = 𝑦 ) ) |
4 |
|
df-rex |
⊢ ( ∃ 𝑤 ∈ 𝑥 𝜑 ↔ ∃ 𝑤 ( 𝑤 ∈ 𝑥 ∧ 𝜑 ) ) |
5 |
4
|
bibi2i |
⊢ ( ( 𝑧 ∈ 𝑦 ↔ ∃ 𝑤 ∈ 𝑥 𝜑 ) ↔ ( 𝑧 ∈ 𝑦 ↔ ∃ 𝑤 ( 𝑤 ∈ 𝑥 ∧ 𝜑 ) ) ) |
6 |
5
|
albii |
⊢ ( ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ ∃ 𝑤 ∈ 𝑥 𝜑 ) ↔ ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ ∃ 𝑤 ( 𝑤 ∈ 𝑥 ∧ 𝜑 ) ) ) |
7 |
6
|
exbii |
⊢ ( ∃ 𝑦 ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ ∃ 𝑤 ∈ 𝑥 𝜑 ) ↔ ∃ 𝑦 ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ ∃ 𝑤 ( 𝑤 ∈ 𝑥 ∧ 𝜑 ) ) ) |
8 |
1 3 7
|
3imtr4i |
⊢ ( ∀ 𝑤 ∃* 𝑧 𝜑 → ∃ 𝑦 ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ ∃ 𝑤 ∈ 𝑥 𝜑 ) ) |