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Mirrors > Home > MPE Home > Th. List > 2eu1v | Structured version Visualization version GIF version |
Description: Double existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one. Version of 2eu1 2734 with 𝑥 and 𝑦 distinct, but not requiring ax-13 2389. (Contributed by NM, 3-Dec-2001.) (Revised by Wolf Lammen, 2-Oct-2023.) |
Ref | Expression |
---|---|
2eu1v | ⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ (∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2eu2ex 2727 | . . . . 5 ⊢ (∃!𝑥∃!𝑦𝜑 → ∃𝑥∃𝑦𝜑) | |
2 | moeu 2667 | . . . . . . . 8 ⊢ (∃*𝑦𝜑 ↔ (∃𝑦𝜑 → ∃!𝑦𝜑)) | |
3 | 2 | albii 1819 | . . . . . . 7 ⊢ (∀𝑥∃*𝑦𝜑 ↔ ∀𝑥(∃𝑦𝜑 → ∃!𝑦𝜑)) |
4 | euim 2700 | . . . . . . 7 ⊢ ((∃𝑥∃𝑦𝜑 ∧ ∀𝑥(∃𝑦𝜑 → ∃!𝑦𝜑)) → (∃!𝑥∃!𝑦𝜑 → ∃!𝑥∃𝑦𝜑)) | |
5 | 3, 4 | sylan2b 595 | . . . . . 6 ⊢ ((∃𝑥∃𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) → (∃!𝑥∃!𝑦𝜑 → ∃!𝑥∃𝑦𝜑)) |
6 | 5 | ex 415 | . . . . 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 | 2euswapv 2714 | . . . 4 ⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃𝑦𝜑 → ∃!𝑦∃𝑥𝜑)) | |
10 | 8, 9 | syld 47 | . . 3 ⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 → ∃!𝑦∃𝑥𝜑)) |
11 | 8, 10 | jcad 515 | . 2 ⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 → (∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑))) |
12 | 2exeuv 2716 | . 2 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) → ∃!𝑥∃!𝑦𝜑) | |
13 | 11, 12 | impbid1 227 | 1 ⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ (∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1534 ∃wex 1779 ∃*wmo 2619 ∃!weu 2652 |
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 1969 ax-7 2014 ax-10 2144 ax-11 2160 ax-12 2176 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 |
This theorem is referenced by: 2eu5 2739 |
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