Step |
Hyp |
Ref |
Expression |
1 |
|
elissetv |
⊢ ( 𝐴 ∈ 𝑉 → ∃ 𝑦 𝑦 = 𝐴 ) |
2 |
|
cbvexeqsetf |
⊢ ( Ⅎ 𝑥 𝐴 → ( ∃ 𝑥 𝑥 = 𝐴 ↔ ∃ 𝑦 𝑦 = 𝐴 ) ) |
3 |
1 2
|
imbitrrid |
⊢ ( Ⅎ 𝑥 𝐴 → ( 𝐴 ∈ 𝑉 → ∃ 𝑥 𝑥 = 𝐴 ) ) |
4 |
|
pm2.04 |
⊢ ( ( 𝑥 = 𝐴 → ( 𝜑 → 𝜓 ) ) → ( 𝜑 → ( 𝑥 = 𝐴 → 𝜓 ) ) ) |
5 |
4
|
al2imi |
⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 → 𝜓 ) ) → ( ∀ 𝑥 𝜑 → ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜓 ) ) ) |
6 |
|
19.23t |
⊢ ( Ⅎ 𝑥 𝜓 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜓 ) ↔ ( ∃ 𝑥 𝑥 = 𝐴 → 𝜓 ) ) ) |
7 |
6
|
biimpd |
⊢ ( Ⅎ 𝑥 𝜓 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜓 ) → ( ∃ 𝑥 𝑥 = 𝐴 → 𝜓 ) ) ) |
8 |
5 7
|
sylan9r |
⊢ ( ( Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 → 𝜓 ) ) ) → ( ∀ 𝑥 𝜑 → ( ∃ 𝑥 𝑥 = 𝐴 → 𝜓 ) ) ) |
9 |
8
|
com23 |
⊢ ( ( Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 → 𝜓 ) ) ) → ( ∃ 𝑥 𝑥 = 𝐴 → ( ∀ 𝑥 𝜑 → 𝜓 ) ) ) |
10 |
3 9
|
sylan9 |
⊢ ( ( Ⅎ 𝑥 𝐴 ∧ ( Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 → 𝜓 ) ) ) ) → ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 𝜑 → 𝜓 ) ) ) |
11 |
10
|
anassrs |
⊢ ( ( ( Ⅎ 𝑥 𝐴 ∧ Ⅎ 𝑥 𝜓 ) ∧ ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 → 𝜓 ) ) ) → ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 𝜑 → 𝜓 ) ) ) |