| Step | Hyp | Ref
| Expression |
| 1 | | euabsn2 4725 |
. . 3
⊢
(∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃𝑧{𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧}) |
| 2 | | eleq2 2830 |
. . . . . 6
⊢ ({𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} → (𝑥 ∈ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} ↔ 𝑥 ∈ {𝑧})) |
| 3 | | abid 2718 |
. . . . . 6
⊢ (𝑥 ∈ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} ↔ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵) |
| 4 | | velsn 4642 |
. . . . . 6
⊢ (𝑥 ∈ {𝑧} ↔ 𝑥 = 𝑧) |
| 5 | 2, 3, 4 | 3bitr3g 313 |
. . . . 5
⊢ ({𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ 𝑥 = 𝑧)) |
| 6 | | nfre1 3285 |
. . . . . . . . 9
⊢
Ⅎ𝑦∃𝑦 ∈ 𝐴 𝑥 = 𝐵 |
| 7 | 6 | nfab 2911 |
. . . . . . . 8
⊢
Ⅎ𝑦{𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} |
| 8 | 7 | nfeq1 2921 |
. . . . . . 7
⊢
Ⅎ𝑦{𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} |
| 9 | | eusvobj1.1 |
. . . . . . . . 9
⊢ 𝐵 ∈ V |
| 10 | 9 | elabrex 7262 |
. . . . . . . 8
⊢ (𝑦 ∈ 𝐴 → 𝐵 ∈ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵}) |
| 11 | | eleq2 2830 |
. . . . . . . . 9
⊢ ({𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} → (𝐵 ∈ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} ↔ 𝐵 ∈ {𝑧})) |
| 12 | 9 | elsn 4641 |
. . . . . . . . . 10
⊢ (𝐵 ∈ {𝑧} ↔ 𝐵 = 𝑧) |
| 13 | | eqcom 2744 |
. . . . . . . . . 10
⊢ (𝐵 = 𝑧 ↔ 𝑧 = 𝐵) |
| 14 | 12, 13 | bitri 275 |
. . . . . . . . 9
⊢ (𝐵 ∈ {𝑧} ↔ 𝑧 = 𝐵) |
| 15 | 11, 14 | bitrdi 287 |
. . . . . . . 8
⊢ ({𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} → (𝐵 ∈ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} ↔ 𝑧 = 𝐵)) |
| 16 | 10, 15 | imbitrid 244 |
. . . . . . 7
⊢ ({𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} → (𝑦 ∈ 𝐴 → 𝑧 = 𝐵)) |
| 17 | 8, 16 | ralrimi 3257 |
. . . . . 6
⊢ ({𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} → ∀𝑦 ∈ 𝐴 𝑧 = 𝐵) |
| 18 | | eqeq1 2741 |
. . . . . . 7
⊢ (𝑥 = 𝑧 → (𝑥 = 𝐵 ↔ 𝑧 = 𝐵)) |
| 19 | 18 | ralbidv 3178 |
. . . . . 6
⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝑧 = 𝐵)) |
| 20 | 17, 19 | syl5ibrcom 247 |
. . . . 5
⊢ ({𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} → (𝑥 = 𝑧 → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵)) |
| 21 | 5, 20 | sylbid 240 |
. . . 4
⊢ ({𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵)) |
| 22 | 21 | exlimiv 1930 |
. . 3
⊢
(∃𝑧{𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵)) |
| 23 | 1, 22 | sylbi 217 |
. 2
⊢
(∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵)) |
| 24 | | euex 2577 |
. . 3
⊢
(∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵) |
| 25 | | rexn0 4511 |
. . . 4
⊢
(∃𝑦 ∈
𝐴 𝑥 = 𝐵 → 𝐴 ≠ ∅) |
| 26 | 25 | exlimiv 1930 |
. . 3
⊢
(∃𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → 𝐴 ≠ ∅) |
| 27 | | r19.2z 4495 |
. . . 4
⊢ ((𝐴 ≠ ∅ ∧
∀𝑦 ∈ 𝐴 𝑥 = 𝐵) → ∃𝑦 ∈ 𝐴 𝑥 = 𝐵) |
| 28 | 27 | ex 412 |
. . 3
⊢ (𝐴 ≠ ∅ →
(∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 = 𝐵)) |
| 29 | 24, 26, 28 | 3syl 18 |
. 2
⊢
(∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → (∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 = 𝐵)) |
| 30 | 23, 29 | impbid 212 |
1
⊢
(∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝑥 = 𝐵)) |