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| Description: Double existential uniqueness implies double unique existential quantification. Version of 2exeu 2645 with 𝑥 and 𝑦 distinct, but not requiring ax-13 2376. (Contributed by NM, 3-Dec-2001.) (Revised by Wolf Lammen, 2-Oct-2023.) | 
| Ref | Expression | 
|---|---|
| 2exeuv | ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) → ∃!𝑥∃!𝑦𝜑) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eumo 2577 | . . . 4 ⊢ (∃!𝑥∃𝑦𝜑 → ∃*𝑥∃𝑦𝜑) | |
| 2 | euex 2576 | . . . . 5 ⊢ (∃!𝑦𝜑 → ∃𝑦𝜑) | |
| 3 | 2 | moimi 2544 | . . . 4 ⊢ (∃*𝑥∃𝑦𝜑 → ∃*𝑥∃!𝑦𝜑) | 
| 4 | 1, 3 | syl 17 | . . 3 ⊢ (∃!𝑥∃𝑦𝜑 → ∃*𝑥∃!𝑦𝜑) | 
| 5 | 2euexv 2630 | . . 3 ⊢ (∃!𝑦∃𝑥𝜑 → ∃𝑥∃!𝑦𝜑) | |
| 6 | 4, 5 | anim12ci 614 | . 2 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) → (∃𝑥∃!𝑦𝜑 ∧ ∃*𝑥∃!𝑦𝜑)) | 
| 7 | df-eu 2568 | . 2 ⊢ (∃!𝑥∃!𝑦𝜑 ↔ (∃𝑥∃!𝑦𝜑 ∧ ∃*𝑥∃!𝑦𝜑)) | |
| 8 | 6, 7 | sylibr 234 | 1 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) → ∃!𝑥∃!𝑦𝜑) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∃wex 1778 ∃*wmo 2537 ∃!weu 2567 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-10 2140 ax-11 2156 ax-12 2176 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-nf 1783 df-mo 2539 df-eu 2568 | 
| This theorem is referenced by: 2eu1v 2651 | 
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