Step |
Hyp |
Ref |
Expression |
1 |
|
elex |
⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ∈ V ) |
2 |
|
issetft |
⊢ ( Ⅎ 𝑥 𝐴 → ( 𝐴 ∈ V ↔ ∃ 𝑥 𝑥 = 𝐴 ) ) |
3 |
1 2
|
imbitrid |
⊢ ( Ⅎ 𝑥 𝐴 → ( 𝐴 ∈ 𝑉 → ∃ 𝑥 𝑥 = 𝐴 ) ) |
4 |
3
|
ad2antrr |
⊢ ( ( ( Ⅎ 𝑥 𝐴 ∧ Ⅎ 𝑥 𝜓 ) ∧ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) ∧ ∀ 𝑥 𝜑 ) ) → ( 𝐴 ∈ 𝑉 → ∃ 𝑥 𝑥 = 𝐴 ) ) |
5 |
4
|
3impia |
⊢ ( ( ( Ⅎ 𝑥 𝐴 ∧ Ⅎ 𝑥 𝜓 ) ∧ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) ∧ ∀ 𝑥 𝜑 ) ∧ 𝐴 ∈ 𝑉 ) → ∃ 𝑥 𝑥 = 𝐴 ) |
6 |
|
biimp |
⊢ ( ( 𝜑 ↔ 𝜓 ) → ( 𝜑 → 𝜓 ) ) |
7 |
6
|
imim2i |
⊢ ( ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) → ( 𝑥 = 𝐴 → ( 𝜑 → 𝜓 ) ) ) |
8 |
7
|
com23 |
⊢ ( ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) → ( 𝜑 → ( 𝑥 = 𝐴 → 𝜓 ) ) ) |
9 |
8
|
imp |
⊢ ( ( ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) ∧ 𝜑 ) → ( 𝑥 = 𝐴 → 𝜓 ) ) |
10 |
9
|
alanimi |
⊢ ( ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) ∧ ∀ 𝑥 𝜑 ) → ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜓 ) ) |
11 |
|
19.23t |
⊢ ( Ⅎ 𝑥 𝜓 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜓 ) ↔ ( ∃ 𝑥 𝑥 = 𝐴 → 𝜓 ) ) ) |
12 |
11
|
adantl |
⊢ ( ( Ⅎ 𝑥 𝐴 ∧ Ⅎ 𝑥 𝜓 ) → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜓 ) ↔ ( ∃ 𝑥 𝑥 = 𝐴 → 𝜓 ) ) ) |
13 |
10 12
|
imbitrid |
⊢ ( ( Ⅎ 𝑥 𝐴 ∧ Ⅎ 𝑥 𝜓 ) → ( ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) ∧ ∀ 𝑥 𝜑 ) → ( ∃ 𝑥 𝑥 = 𝐴 → 𝜓 ) ) ) |
14 |
13
|
imp |
⊢ ( ( ( Ⅎ 𝑥 𝐴 ∧ Ⅎ 𝑥 𝜓 ) ∧ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) ∧ ∀ 𝑥 𝜑 ) ) → ( ∃ 𝑥 𝑥 = 𝐴 → 𝜓 ) ) |
15 |
14
|
3adant3 |
⊢ ( ( ( Ⅎ 𝑥 𝐴 ∧ Ⅎ 𝑥 𝜓 ) ∧ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) ∧ ∀ 𝑥 𝜑 ) ∧ 𝐴 ∈ 𝑉 ) → ( ∃ 𝑥 𝑥 = 𝐴 → 𝜓 ) ) |
16 |
5 15
|
mpd |
⊢ ( ( ( Ⅎ 𝑥 𝐴 ∧ Ⅎ 𝑥 𝜓 ) ∧ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) ∧ ∀ 𝑥 𝜑 ) ∧ 𝐴 ∈ 𝑉 ) → 𝜓 ) |