| Step | Hyp | Ref
| Expression |
| 1 | | ax-ext 2708 |
. . . . . 6
⊢
(∀𝑤(𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑧) → 𝑥 = 𝑧) |
| 2 | 1 | alimi 1811 |
. . . . 5
⊢
(∀𝑥∀𝑤(𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑧) → ∀𝑥 𝑥 = 𝑧) |
| 3 | | ax-11 2157 |
. . . . . . 7
⊢
(∀𝑥∀𝑤(𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑧) → ∀𝑤∀𝑥(𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑧)) |
| 4 | | ax9 2122 |
. . . . . . . . 9
⊢ (𝑥 = 𝑧 → (𝑤 ∈ 𝑥 → 𝑤 ∈ 𝑧)) |
| 5 | | biimpr 220 |
. . . . . . . . . . 11
⊢ ((𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑧) → (𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥)) |
| 6 | 5 | alimi 1811 |
. . . . . . . . . 10
⊢
(∀𝑥(𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑧) → ∀𝑥(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥)) |
| 7 | | stdpc5v 1938 |
. . . . . . . . . 10
⊢
(∀𝑥(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → (𝑤 ∈ 𝑧 → ∀𝑥 𝑤 ∈ 𝑥)) |
| 8 | 6, 7 | syl 17 |
. . . . . . . . 9
⊢
(∀𝑥(𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑧) → (𝑤 ∈ 𝑧 → ∀𝑥 𝑤 ∈ 𝑥)) |
| 9 | 4, 8 | syl9 77 |
. . . . . . . 8
⊢ (𝑥 = 𝑧 → (∀𝑥(𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑧) → (𝑤 ∈ 𝑥 → ∀𝑥 𝑤 ∈ 𝑥))) |
| 10 | 9 | alimdv 1916 |
. . . . . . 7
⊢ (𝑥 = 𝑧 → (∀𝑤∀𝑥(𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑧) → ∀𝑤(𝑤 ∈ 𝑥 → ∀𝑥 𝑤 ∈ 𝑥))) |
| 11 | 3, 10 | syl5 34 |
. . . . . 6
⊢ (𝑥 = 𝑧 → (∀𝑥∀𝑤(𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑧) → ∀𝑤(𝑤 ∈ 𝑥 → ∀𝑥 𝑤 ∈ 𝑥))) |
| 12 | 11 | sps 2185 |
. . . . 5
⊢
(∀𝑥 𝑥 = 𝑧 → (∀𝑥∀𝑤(𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑧) → ∀𝑤(𝑤 ∈ 𝑥 → ∀𝑥 𝑤 ∈ 𝑥))) |
| 13 | 2, 12 | mpcom 38 |
. . . 4
⊢
(∀𝑥∀𝑤(𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑧) → ∀𝑤(𝑤 ∈ 𝑥 → ∀𝑥 𝑤 ∈ 𝑥)) |
| 14 | 13 | axc4i 2322 |
. . 3
⊢
(∀𝑥∀𝑤(𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑧) → ∀𝑥∀𝑤(𝑤 ∈ 𝑥 → ∀𝑥 𝑤 ∈ 𝑥)) |
| 15 | | nfa1 2151 |
. . . . . . . 8
⊢
Ⅎ𝑥∀𝑥 𝑤 ∈ 𝑥 |
| 16 | 15 | 19.23 2211 |
. . . . . . 7
⊢
(∀𝑥(𝑤 ∈ 𝑥 → ∀𝑥 𝑤 ∈ 𝑥) ↔ (∃𝑥 𝑤 ∈ 𝑥 → ∀𝑥 𝑤 ∈ 𝑥)) |
| 17 | | 19.8a 2181 |
. . . . . . . . 9
⊢ (𝑤 ∈ 𝑧 → ∃𝑧 𝑤 ∈ 𝑧) |
| 18 | | elequ2 2123 |
. . . . . . . . . 10
⊢ (𝑧 = 𝑥 → (𝑤 ∈ 𝑧 ↔ 𝑤 ∈ 𝑥)) |
| 19 | 18 | cbvexvw 2036 |
. . . . . . . . 9
⊢
(∃𝑧 𝑤 ∈ 𝑧 ↔ ∃𝑥 𝑤 ∈ 𝑥) |
| 20 | 17, 19 | sylib 218 |
. . . . . . . 8
⊢ (𝑤 ∈ 𝑧 → ∃𝑥 𝑤 ∈ 𝑥) |
| 21 | 4 | cbvalivw 2006 |
. . . . . . . 8
⊢
(∀𝑥 𝑤 ∈ 𝑥 → ∀𝑧 𝑤 ∈ 𝑧) |
| 22 | 20, 21 | imim12i 62 |
. . . . . . 7
⊢
((∃𝑥 𝑤 ∈ 𝑥 → ∀𝑥 𝑤 ∈ 𝑥) → (𝑤 ∈ 𝑧 → ∀𝑧 𝑤 ∈ 𝑧)) |
| 23 | 16, 22 | sylbi 217 |
. . . . . 6
⊢
(∀𝑥(𝑤 ∈ 𝑥 → ∀𝑥 𝑤 ∈ 𝑥) → (𝑤 ∈ 𝑧 → ∀𝑧 𝑤 ∈ 𝑧)) |
| 24 | 23 | alimi 1811 |
. . . . 5
⊢
(∀𝑤∀𝑥(𝑤 ∈ 𝑥 → ∀𝑥 𝑤 ∈ 𝑥) → ∀𝑤(𝑤 ∈ 𝑧 → ∀𝑧 𝑤 ∈ 𝑧)) |
| 25 | 24 | alcoms 2158 |
. . . 4
⊢
(∀𝑥∀𝑤(𝑤 ∈ 𝑥 → ∀𝑥 𝑤 ∈ 𝑥) → ∀𝑤(𝑤 ∈ 𝑧 → ∀𝑧 𝑤 ∈ 𝑧)) |
| 26 | 25 | alrimiv 1927 |
. . 3
⊢
(∀𝑥∀𝑤(𝑤 ∈ 𝑥 → ∀𝑥 𝑤 ∈ 𝑥) → ∀𝑧∀𝑤(𝑤 ∈ 𝑧 → ∀𝑧 𝑤 ∈ 𝑧)) |
| 27 | | nfa1 2151 |
. . . . . . . 8
⊢
Ⅎ𝑧∀𝑧 𝑤 ∈ 𝑧 |
| 28 | 27 | 19.23 2211 |
. . . . . . 7
⊢
(∀𝑧(𝑤 ∈ 𝑧 → ∀𝑧 𝑤 ∈ 𝑧) ↔ (∃𝑧 𝑤 ∈ 𝑧 → ∀𝑧 𝑤 ∈ 𝑧)) |
| 29 | | ax9 2122 |
. . . . . . . . . 10
⊢ (𝑧 = 𝑥 → (𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥)) |
| 30 | 29 | spimvw 1995 |
. . . . . . . . 9
⊢
(∀𝑧 𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) |
| 31 | 17, 30 | imim12i 62 |
. . . . . . . 8
⊢
((∃𝑧 𝑤 ∈ 𝑧 → ∀𝑧 𝑤 ∈ 𝑧) → (𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥)) |
| 32 | | 19.8a 2181 |
. . . . . . . . . 10
⊢ (𝑤 ∈ 𝑥 → ∃𝑥 𝑤 ∈ 𝑥) |
| 33 | | elequ2 2123 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑧 → (𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑧)) |
| 34 | 33 | cbvexvw 2036 |
. . . . . . . . . 10
⊢
(∃𝑥 𝑤 ∈ 𝑥 ↔ ∃𝑧 𝑤 ∈ 𝑧) |
| 35 | 32, 34 | sylib 218 |
. . . . . . . . 9
⊢ (𝑤 ∈ 𝑥 → ∃𝑧 𝑤 ∈ 𝑧) |
| 36 | | sp 2183 |
. . . . . . . . 9
⊢
(∀𝑧 𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑧) |
| 37 | 35, 36 | imim12i 62 |
. . . . . . . 8
⊢
((∃𝑧 𝑤 ∈ 𝑧 → ∀𝑧 𝑤 ∈ 𝑧) → (𝑤 ∈ 𝑥 → 𝑤 ∈ 𝑧)) |
| 38 | 31, 37 | impbid 212 |
. . . . . . 7
⊢
((∃𝑧 𝑤 ∈ 𝑧 → ∀𝑧 𝑤 ∈ 𝑧) → (𝑤 ∈ 𝑧 ↔ 𝑤 ∈ 𝑥)) |
| 39 | 28, 38 | sylbi 217 |
. . . . . 6
⊢
(∀𝑧(𝑤 ∈ 𝑧 → ∀𝑧 𝑤 ∈ 𝑧) → (𝑤 ∈ 𝑧 ↔ 𝑤 ∈ 𝑥)) |
| 40 | 39 | alimi 1811 |
. . . . 5
⊢
(∀𝑤∀𝑧(𝑤 ∈ 𝑧 → ∀𝑧 𝑤 ∈ 𝑧) → ∀𝑤(𝑤 ∈ 𝑧 ↔ 𝑤 ∈ 𝑥)) |
| 41 | 40 | alcoms 2158 |
. . . 4
⊢
(∀𝑧∀𝑤(𝑤 ∈ 𝑧 → ∀𝑧 𝑤 ∈ 𝑧) → ∀𝑤(𝑤 ∈ 𝑧 ↔ 𝑤 ∈ 𝑥)) |
| 42 | 41 | axc4i 2322 |
. . 3
⊢
(∀𝑧∀𝑤(𝑤 ∈ 𝑧 → ∀𝑧 𝑤 ∈ 𝑧) → ∀𝑧∀𝑤(𝑤 ∈ 𝑧 ↔ 𝑤 ∈ 𝑥)) |
| 43 | 14, 26, 42 | 3syl 18 |
. 2
⊢
(∀𝑥∀𝑤(𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑧) → ∀𝑧∀𝑤(𝑤 ∈ 𝑧 ↔ 𝑤 ∈ 𝑥)) |
| 44 | | axextb 2711 |
. . 3
⊢ (𝑥 = 𝑧 ↔ ∀𝑤(𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑧)) |
| 45 | 44 | albii 1819 |
. 2
⊢
(∀𝑥 𝑥 = 𝑧 ↔ ∀𝑥∀𝑤(𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑧)) |
| 46 | | axextb 2711 |
. . 3
⊢ (𝑧 = 𝑥 ↔ ∀𝑤(𝑤 ∈ 𝑧 ↔ 𝑤 ∈ 𝑥)) |
| 47 | 46 | albii 1819 |
. 2
⊢
(∀𝑧 𝑧 = 𝑥 ↔ ∀𝑧∀𝑤(𝑤 ∈ 𝑧 ↔ 𝑤 ∈ 𝑥)) |
| 48 | 43, 45, 47 | 3imtr4i 292 |
1
⊢
(∀𝑥 𝑥 = 𝑧 → ∀𝑧 𝑧 = 𝑥) |