Step |
Hyp |
Ref |
Expression |
1 |
|
bj-ax12 |
⊢ ∀ 𝑥 ( 𝑥 = 𝑦 → ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) |
2 |
|
pm3.31 |
⊢ ( ( 𝑥 = 𝑦 → ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) → ( ( 𝑥 = 𝑦 ∧ 𝜑 ) → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) |
3 |
2
|
aleximi |
⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) → ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) → ∃ 𝑥 ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) |
4 |
1 3
|
ax-mp |
⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) → ∃ 𝑥 ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) |
5 |
|
hbe1a |
⊢ ( ∃ 𝑥 ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) |
6 |
4 5
|
syl |
⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) |
7 |
|
equs4v |
⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ) |
8 |
6 7
|
impbii |
⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) |