Step |
Hyp |
Ref |
Expression |
1 |
|
dvelimexcased.1 |
⊢ Ⅎ 𝑥 𝜑 |
2 |
|
dvelimexcased.2 |
⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑧 𝜑 ) |
3 |
|
dvelimexcased.3 |
⊢ ( ( 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → Ⅎ 𝑥 𝜓 ) |
4 |
|
dvelimexcased.4 |
⊢ ( ( 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → Ⅎ 𝑧 𝜃 ) |
5 |
|
dvelimexcased.5 |
⊢ ( ( 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → ( 𝑧 = 𝑥 → ( 𝜓 → 𝜃 ) ) ) |
6 |
|
dvelimexcased.6 |
⊢ ( ( 𝜑 ∧ ∀ 𝑥 𝑥 = 𝑦 ) → ( 𝜒 → 𝜃 ) ) |
7 |
|
dvelimexcased.7 |
⊢ ( 𝜑 → ∃ 𝑧 𝜓 ) |
8 |
|
dvelimexcased.8 |
⊢ ( 𝜑 → ∃ 𝑥 𝜒 ) |
9 |
|
nfa1 |
⊢ Ⅎ 𝑥 ∀ 𝑥 𝑥 = 𝑦 |
10 |
1 9
|
nfan |
⊢ Ⅎ 𝑥 ( 𝜑 ∧ ∀ 𝑥 𝑥 = 𝑦 ) |
11 |
10 6
|
eximd |
⊢ ( ( 𝜑 ∧ ∀ 𝑥 𝑥 = 𝑦 ) → ( ∃ 𝑥 𝜒 → ∃ 𝑥 𝜃 ) ) |
12 |
11
|
ex |
⊢ ( 𝜑 → ( ∀ 𝑥 𝑥 = 𝑦 → ( ∃ 𝑥 𝜒 → ∃ 𝑥 𝜃 ) ) ) |
13 |
8 12
|
mpid |
⊢ ( 𝜑 → ( ∀ 𝑥 𝑥 = 𝑦 → ∃ 𝑥 𝜃 ) ) |
14 |
|
nfv |
⊢ Ⅎ 𝑧 ¬ ∀ 𝑥 𝑥 = 𝑦 |
15 |
14 2
|
nfan1c |
⊢ Ⅎ 𝑧 ( 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) |
16 |
|
nfna1 |
⊢ Ⅎ 𝑥 ¬ ∀ 𝑥 𝑥 = 𝑦 |
17 |
1 16
|
nfan |
⊢ Ⅎ 𝑥 ( 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) |
18 |
15 17 3 4 5
|
cbvex1v |
⊢ ( ( 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → ( ∃ 𝑧 𝜓 → ∃ 𝑥 𝜃 ) ) |
19 |
18
|
ex |
⊢ ( 𝜑 → ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( ∃ 𝑧 𝜓 → ∃ 𝑥 𝜃 ) ) ) |
20 |
7 19
|
mpid |
⊢ ( 𝜑 → ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ∃ 𝑥 𝜃 ) ) |
21 |
13 20
|
pm2.61d |
⊢ ( 𝜑 → ∃ 𝑥 𝜃 ) |