| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | vex 3483 | . . . . . . 7
⊢ 𝑢 ∈ V | 
| 2 |  | ax6e 2387 | . . . . . . 7
⊢
∃𝑦 𝑦 = 𝑣 | 
| 3 | 1, 2 | pm3.2i 470 | . . . . . 6
⊢ (𝑢 ∈ V ∧ ∃𝑦 𝑦 = 𝑣) | 
| 4 |  | 19.42v 1952 | . . . . . . 7
⊢
(∃𝑦(𝑢 ∈ V ∧ 𝑦 = 𝑣) ↔ (𝑢 ∈ V ∧ ∃𝑦 𝑦 = 𝑣)) | 
| 5 | 4 | biimpri 228 | . . . . . 6
⊢ ((𝑢 ∈ V ∧ ∃𝑦 𝑦 = 𝑣) → ∃𝑦(𝑢 ∈ V ∧ 𝑦 = 𝑣)) | 
| 6 | 3, 5 | ax-mp 5 | . . . . 5
⊢
∃𝑦(𝑢 ∈ V ∧ 𝑦 = 𝑣) | 
| 7 |  | isset 3493 | . . . . . . 7
⊢ (𝑢 ∈ V ↔ ∃𝑥 𝑥 = 𝑢) | 
| 8 | 7 | anbi1i 624 | . . . . . 6
⊢ ((𝑢 ∈ V ∧ 𝑦 = 𝑣) ↔ (∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) | 
| 9 | 8 | exbii 1847 | . . . . 5
⊢
(∃𝑦(𝑢 ∈ V ∧ 𝑦 = 𝑣) ↔ ∃𝑦(∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) | 
| 10 | 6, 9 | mpbi 230 | . . . 4
⊢
∃𝑦(∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣) | 
| 11 |  | id 22 | . . . . . 6
⊢ (¬
∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑥 = 𝑦) | 
| 12 |  | hbnae 2436 | . . . . . . 7
⊢ (¬
∀𝑥 𝑥 = 𝑦 → ∀𝑦 ¬ ∀𝑥 𝑥 = 𝑦) | 
| 13 |  | hbn1 2141 | . . . . . . . . . . . 12
⊢ (¬
∀𝑥 𝑥 = 𝑦 → ∀𝑥 ¬ ∀𝑥 𝑥 = 𝑦) | 
| 14 |  | ax-5 1909 | . . . . . . . . . . . . . . . 16
⊢ (𝑧 = 𝑣 → ∀𝑥 𝑧 = 𝑣) | 
| 15 |  | ax-5 1909 | . . . . . . . . . . . . . . . 16
⊢ (𝑦 = 𝑣 → ∀𝑧 𝑦 = 𝑣) | 
| 16 |  | id 22 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑧 = 𝑦 → 𝑧 = 𝑦) | 
| 17 |  | equequ1 2023 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑧 = 𝑦 → (𝑧 = 𝑣 ↔ 𝑦 = 𝑣)) | 
| 18 | 16, 17 | syl 17 | . . . . . . . . . . . . . . . . 17
⊢ (𝑧 = 𝑦 → (𝑧 = 𝑣 ↔ 𝑦 = 𝑣)) | 
| 19 | 18 | idiALT 44503 | . . . . . . . . . . . . . . . 16
⊢ (𝑧 = 𝑦 → (𝑧 = 𝑣 ↔ 𝑦 = 𝑣)) | 
| 20 | 14, 15, 19 | dvelimh 2454 | . . . . . . . . . . . . . . 15
⊢ (¬
∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑣 → ∀𝑥 𝑦 = 𝑣)) | 
| 21 | 11, 20 | syl 17 | . . . . . . . . . . . . . 14
⊢ (¬
∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑣 → ∀𝑥 𝑦 = 𝑣)) | 
| 22 | 21 | idiALT 44503 | . . . . . . . . . . . . 13
⊢ (¬
∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑣 → ∀𝑥 𝑦 = 𝑣)) | 
| 23 | 22 | alimi 1810 | . . . . . . . . . . . 12
⊢
(∀𝑥 ¬
∀𝑥 𝑥 = 𝑦 → ∀𝑥(𝑦 = 𝑣 → ∀𝑥 𝑦 = 𝑣)) | 
| 24 | 13, 23 | syl 17 | . . . . . . . . . . 11
⊢ (¬
∀𝑥 𝑥 = 𝑦 → ∀𝑥(𝑦 = 𝑣 → ∀𝑥 𝑦 = 𝑣)) | 
| 25 | 11, 24 | syl 17 | . . . . . . . . . 10
⊢ (¬
∀𝑥 𝑥 = 𝑦 → ∀𝑥(𝑦 = 𝑣 → ∀𝑥 𝑦 = 𝑣)) | 
| 26 |  | 19.41rg 44575 | . . . . . . . . . 10
⊢
(∀𝑥(𝑦 = 𝑣 → ∀𝑥 𝑦 = 𝑣) → ((∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))) | 
| 27 | 25, 26 | syl 17 | . . . . . . . . 9
⊢ (¬
∀𝑥 𝑥 = 𝑦 → ((∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))) | 
| 28 | 27 | idiALT 44503 | . . . . . . . 8
⊢ (¬
∀𝑥 𝑥 = 𝑦 → ((∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))) | 
| 29 | 28 | alimi 1810 | . . . . . . 7
⊢
(∀𝑦 ¬
∀𝑥 𝑥 = 𝑦 → ∀𝑦((∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))) | 
| 30 | 12, 29 | syl 17 | . . . . . 6
⊢ (¬
∀𝑥 𝑥 = 𝑦 → ∀𝑦((∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))) | 
| 31 | 11, 30 | syl 17 | . . . . 5
⊢ (¬
∀𝑥 𝑥 = 𝑦 → ∀𝑦((∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))) | 
| 32 |  | exim 1833 | . . . . 5
⊢
(∀𝑦((∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) → (∃𝑦(∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))) | 
| 33 | 31, 32 | syl 17 | . . . 4
⊢ (¬
∀𝑥 𝑥 = 𝑦 → (∃𝑦(∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))) | 
| 34 |  | pm2.27 42 | . . . 4
⊢
(∃𝑦(∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ((∃𝑦(∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) → ∃𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))) | 
| 35 | 10, 33, 34 | mpsyl 68 | . . 3
⊢ (¬
∀𝑥 𝑥 = 𝑦 → ∃𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) | 
| 36 |  | excomim 2162 | . . 3
⊢
(∃𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) | 
| 37 | 35, 36 | syl 17 | . 2
⊢ (¬
∀𝑥 𝑥 = 𝑦 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) | 
| 38 | 37 | idiALT 44503 | 1
⊢ (¬
∀𝑥 𝑥 = 𝑦 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) |