| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | axpowndlem4 10641 | . 2
⊢ (¬
∀𝑦 𝑦 = 𝑥 → (¬ ∀𝑦 𝑦 = 𝑧 → (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))) | 
| 2 |  | axpowndlem1 10638 | . . 3
⊢
(∀𝑥 𝑥 = 𝑦 → (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))) | 
| 3 | 2 | aecoms 2432 | . 2
⊢
(∀𝑦 𝑦 = 𝑥 → (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))) | 
| 4 | 2 | a1d 25 | . . 3
⊢
(∀𝑥 𝑥 = 𝑦 → (∀𝑦 𝑦 = 𝑧 → (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))) | 
| 5 |  | nfnae 2438 | . . . . . . . 8
⊢
Ⅎ𝑦 ¬
∀𝑥 𝑥 = 𝑦 | 
| 6 |  | nfae 2437 | . . . . . . . 8
⊢
Ⅎ𝑦∀𝑦 𝑦 = 𝑧 | 
| 7 | 5, 6 | nfan 1898 | . . . . . . 7
⊢
Ⅎ𝑦(¬
∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) | 
| 8 |  | el 5441 | . . . . . . . . . . . . 13
⊢
∃𝑤 𝑥 ∈ 𝑤 | 
| 9 |  | nfcvf2 2932 | . . . . . . . . . . . . . . 15
⊢ (¬
∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑥) | 
| 10 |  | nfcvd 2905 | . . . . . . . . . . . . . . 15
⊢ (¬
∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑤) | 
| 11 | 9, 10 | nfeld 2916 | . . . . . . . . . . . . . 14
⊢ (¬
∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 𝑥 ∈ 𝑤) | 
| 12 |  | elequ2 2122 | . . . . . . . . . . . . . . 15
⊢ (𝑤 = 𝑦 → (𝑥 ∈ 𝑤 ↔ 𝑥 ∈ 𝑦)) | 
| 13 | 12 | a1i 11 | . . . . . . . . . . . . . 14
⊢ (¬
∀𝑥 𝑥 = 𝑦 → (𝑤 = 𝑦 → (𝑥 ∈ 𝑤 ↔ 𝑥 ∈ 𝑦))) | 
| 14 | 5, 11, 13 | cbvexd 2412 | . . . . . . . . . . . . 13
⊢ (¬
∀𝑥 𝑥 = 𝑦 → (∃𝑤 𝑥 ∈ 𝑤 ↔ ∃𝑦 𝑥 ∈ 𝑦)) | 
| 15 | 8, 14 | mpbii 233 | . . . . . . . . . . . 12
⊢ (¬
∀𝑥 𝑥 = 𝑦 → ∃𝑦 𝑥 ∈ 𝑦) | 
| 16 | 15 | 19.8ad 2181 | . . . . . . . . . . 11
⊢ (¬
∀𝑥 𝑥 = 𝑦 → ∃𝑥∃𝑦 𝑥 ∈ 𝑦) | 
| 17 |  | df-ex 1779 | . . . . . . . . . . 11
⊢
(∃𝑥∃𝑦 𝑥 ∈ 𝑦 ↔ ¬ ∀𝑥 ¬ ∃𝑦 𝑥 ∈ 𝑦) | 
| 18 | 16, 17 | sylib 218 | . . . . . . . . . 10
⊢ (¬
∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 ¬ ∃𝑦 𝑥 ∈ 𝑦) | 
| 19 | 18 | adantr 480 | . . . . . . . . 9
⊢ ((¬
∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → ¬ ∀𝑥 ¬ ∃𝑦 𝑥 ∈ 𝑦) | 
| 20 |  | biidd 262 | . . . . . . . . . . . . . 14
⊢
(∀𝑦 𝑦 = 𝑧 → (¬ 𝑥 ∈ 𝑦 ↔ ¬ 𝑥 ∈ 𝑦)) | 
| 21 | 20 | dral1 2443 | . . . . . . . . . . . . 13
⊢
(∀𝑦 𝑦 = 𝑧 → (∀𝑦 ¬ 𝑥 ∈ 𝑦 ↔ ∀𝑧 ¬ 𝑥 ∈ 𝑦)) | 
| 22 |  | alnex 1780 | . . . . . . . . . . . . 13
⊢
(∀𝑦 ¬
𝑥 ∈ 𝑦 ↔ ¬ ∃𝑦 𝑥 ∈ 𝑦) | 
| 23 |  | alnex 1780 | . . . . . . . . . . . . 13
⊢
(∀𝑧 ¬
𝑥 ∈ 𝑦 ↔ ¬ ∃𝑧 𝑥 ∈ 𝑦) | 
| 24 | 21, 22, 23 | 3bitr3g 313 | . . . . . . . . . . . 12
⊢
(∀𝑦 𝑦 = 𝑧 → (¬ ∃𝑦 𝑥 ∈ 𝑦 ↔ ¬ ∃𝑧 𝑥 ∈ 𝑦)) | 
| 25 |  | nd2 10629 | . . . . . . . . . . . . 13
⊢
(∀𝑦 𝑦 = 𝑧 → ¬ ∀𝑦 𝑥 ∈ 𝑧) | 
| 26 |  | mtt 364 | . . . . . . . . . . . . 13
⊢ (¬
∀𝑦 𝑥 ∈ 𝑧 → (¬ ∃𝑧 𝑥 ∈ 𝑦 ↔ (∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧))) | 
| 27 | 25, 26 | syl 17 | . . . . . . . . . . . 12
⊢
(∀𝑦 𝑦 = 𝑧 → (¬ ∃𝑧 𝑥 ∈ 𝑦 ↔ (∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧))) | 
| 28 | 24, 27 | bitrd 279 | . . . . . . . . . . 11
⊢
(∀𝑦 𝑦 = 𝑧 → (¬ ∃𝑦 𝑥 ∈ 𝑦 ↔ (∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧))) | 
| 29 | 28 | dral2 2442 | . . . . . . . . . 10
⊢
(∀𝑦 𝑦 = 𝑧 → (∀𝑥 ¬ ∃𝑦 𝑥 ∈ 𝑦 ↔ ∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧))) | 
| 30 | 29 | adantl 481 | . . . . . . . . 9
⊢ ((¬
∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → (∀𝑥 ¬ ∃𝑦 𝑥 ∈ 𝑦 ↔ ∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧))) | 
| 31 | 19, 30 | mtbid 324 | . . . . . . . 8
⊢ ((¬
∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → ¬ ∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧)) | 
| 32 | 31 | pm2.21d 121 | . . . . . . 7
⊢ ((¬
∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → (∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)) | 
| 33 | 7, 32 | alrimi 2212 | . . . . . 6
⊢ ((¬
∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → ∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)) | 
| 34 | 33 | 19.8ad 2181 | . . . . 5
⊢ ((¬
∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)) | 
| 35 | 34 | a1d 25 | . . . 4
⊢ ((¬
∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))) | 
| 36 | 35 | ex 412 | . . 3
⊢ (¬
∀𝑥 𝑥 = 𝑦 → (∀𝑦 𝑦 = 𝑧 → (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))) | 
| 37 | 4, 36 | pm2.61i 182 | . 2
⊢
(∀𝑦 𝑦 = 𝑧 → (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))) | 
| 38 | 1, 3, 37 | pm2.61ii 183 | 1
⊢ (¬
𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)) |