Proof of Theorem dfmoeu
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | alnex 1780 | . . . . 5
⊢
(∀𝑥 ¬
𝜑 ↔ ¬ ∃𝑥𝜑) | 
| 2 |  | pm2.21 123 | . . . . . 6
⊢ (¬
𝜑 → (𝜑 → 𝑥 = 𝑦)) | 
| 3 | 2 | alimi 1810 | . . . . 5
⊢
(∀𝑥 ¬
𝜑 → ∀𝑥(𝜑 → 𝑥 = 𝑦)) | 
| 4 | 1, 3 | sylbir 235 | . . . 4
⊢ (¬
∃𝑥𝜑 → ∀𝑥(𝜑 → 𝑥 = 𝑦)) | 
| 5 | 4 | 19.8ad 2181 | . . 3
⊢ (¬
∃𝑥𝜑 → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) | 
| 6 |  | biimp 215 | . . . . 5
⊢ ((𝜑 ↔ 𝑥 = 𝑦) → (𝜑 → 𝑥 = 𝑦)) | 
| 7 | 6 | alimi 1810 | . . . 4
⊢
(∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∀𝑥(𝜑 → 𝑥 = 𝑦)) | 
| 8 | 7 | eximi 1834 | . . 3
⊢
(∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) | 
| 9 | 5, 8 | ja 186 | . 2
⊢
((∃𝑥𝜑 → ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) | 
| 10 |  | nfia1 2152 | . . . . 5
⊢
Ⅎ𝑥(∀𝑥(𝜑 → 𝑥 = 𝑦) → ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) | 
| 11 |  | id 22 | . . . . . . . . 9
⊢ (𝜑 → 𝜑) | 
| 12 |  | ax12v 2177 | . . . . . . . . . 10
⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | 
| 13 | 12 | com12 32 | . . . . . . . . 9
⊢ (𝜑 → (𝑥 = 𝑦 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | 
| 14 | 11, 13 | embantd 59 | . . . . . . . 8
⊢ (𝜑 → ((𝜑 → 𝑥 = 𝑦) → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | 
| 15 | 14 | spsd 2186 | . . . . . . 7
⊢ (𝜑 → (∀𝑥(𝜑 → 𝑥 = 𝑦) → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | 
| 16 | 15 | ancld 550 | . . . . . 6
⊢ (𝜑 → (∀𝑥(𝜑 → 𝑥 = 𝑦) → (∀𝑥(𝜑 → 𝑥 = 𝑦) ∧ ∀𝑥(𝑥 = 𝑦 → 𝜑)))) | 
| 17 |  | albiim 1888 | . . . . . 6
⊢
(∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ (∀𝑥(𝜑 → 𝑥 = 𝑦) ∧ ∀𝑥(𝑥 = 𝑦 → 𝜑))) | 
| 18 | 16, 17 | imbitrrdi 252 | . . . . 5
⊢ (𝜑 → (∀𝑥(𝜑 → 𝑥 = 𝑦) → ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))) | 
| 19 | 10, 18 | exlimi 2216 | . . . 4
⊢
(∃𝑥𝜑 → (∀𝑥(𝜑 → 𝑥 = 𝑦) → ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))) | 
| 20 | 19 | eximdv 1916 | . . 3
⊢
(∃𝑥𝜑 → (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))) | 
| 21 | 20 | com12 32 | . 2
⊢
(∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → (∃𝑥𝜑 → ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))) | 
| 22 | 9, 21 | impbii 209 | 1
⊢
((∃𝑥𝜑 → ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) |