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| Mirrors > Home > MPE Home > Th. List > 2eu1 | Structured version Visualization version GIF version | ||
| Description: Double existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one. Usage of this theorem is discouraged because it depends on ax-13 2377. Use the weaker 2eu1v 2652 when possible. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 23-Apr-2023.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| 2eu1 | ⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ (∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 2eu2ex 2643 | . . . . 5 ⊢ (∃!𝑥∃!𝑦𝜑 → ∃𝑥∃𝑦𝜑) | |
| 2 | moeu 2583 | . . . . . . . 8 ⊢ (∃*𝑦𝜑 ↔ (∃𝑦𝜑 → ∃!𝑦𝜑)) | |
| 3 | 2 | albii 1819 | . . . . . . 7 ⊢ (∀𝑥∃*𝑦𝜑 ↔ ∀𝑥(∃𝑦𝜑 → ∃!𝑦𝜑)) |
| 4 | euim 2617 | . . . . . . 7 ⊢ ((∃𝑥∃𝑦𝜑 ∧ ∀𝑥(∃𝑦𝜑 → ∃!𝑦𝜑)) → (∃!𝑥∃!𝑦𝜑 → ∃!𝑥∃𝑦𝜑)) | |
| 5 | 3, 4 | sylan2b 594 | . . . . . 6 ⊢ ((∃𝑥∃𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) → (∃!𝑥∃!𝑦𝜑 → ∃!𝑥∃𝑦𝜑)) |
| 6 | 5 | ex 412 | . . . . 5 ⊢ (∃𝑥∃𝑦𝜑 → (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 → ∃!𝑥∃𝑦𝜑))) |
| 7 | 1, 6 | syl 17 | . . . 4 ⊢ (∃!𝑥∃!𝑦𝜑 → (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 → ∃!𝑥∃𝑦𝜑))) |
| 8 | 7 | pm2.43b 55 | . . 3 ⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 → ∃!𝑥∃𝑦𝜑)) |
| 9 | 2euswap 2645 | . . . 4 ⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃𝑦𝜑 → ∃!𝑦∃𝑥𝜑)) | |
| 10 | 8, 9 | syld 47 | . . 3 ⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 → ∃!𝑦∃𝑥𝜑)) |
| 11 | 8, 10 | jcad 512 | . 2 ⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 → (∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑))) |
| 12 | 2exeu 2646 | . 2 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) → ∃!𝑥∃!𝑦𝜑) | |
| 13 | 11, 12 | impbid1 225 | 1 ⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ (∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1538 ∃wex 1779 ∃*wmo 2538 ∃!weu 2568 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-10 2141 ax-11 2157 ax-12 2177 ax-13 2377 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-mo 2540 df-eu 2569 |
| This theorem is referenced by: 2eu2 2653 2eu3 2654 |
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