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Mirrors > Home > MPE Home > Th. List > 2exeu | Structured version Visualization version GIF version |
Description: Double existential uniqueness implies double uniqueness quantification. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) |
Ref | Expression |
---|---|
2exeu | ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) → ∃!𝑥∃!𝑦𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eumo 2647 | . . . 4 ⊢ (∃!𝑥∃𝑦𝜑 → ∃*𝑥∃𝑦𝜑) | |
2 | euex 2642 | . . . . 5 ⊢ (∃!𝑦𝜑 → ∃𝑦𝜑) | |
3 | 2 | moimi 2669 | . . . 4 ⊢ (∃*𝑥∃𝑦𝜑 → ∃*𝑥∃!𝑦𝜑) |
4 | 1, 3 | syl 17 | . . 3 ⊢ (∃!𝑥∃𝑦𝜑 → ∃*𝑥∃!𝑦𝜑) |
5 | 2euex 2693 | . . 3 ⊢ (∃!𝑦∃𝑥𝜑 → ∃𝑥∃!𝑦𝜑) | |
6 | 4, 5 | anim12ci 601 | . 2 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) → (∃𝑥∃!𝑦𝜑 ∧ ∃*𝑥∃!𝑦𝜑)) |
7 | eu5 2644 | . 2 ⊢ (∃!𝑥∃!𝑦𝜑 ↔ (∃𝑥∃!𝑦𝜑 ∧ ∃*𝑥∃!𝑦𝜑)) | |
8 | 6, 7 | sylibr 224 | 1 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) → ∃!𝑥∃!𝑦𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 ∃wex 1852 ∃!weu 2618 ∃*wmo 2619 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-tru 1634 df-ex 1853 df-nf 1858 df-eu 2622 df-mo 2623 |
This theorem is referenced by: 2eu1 2702 2eu2 2703 2eu3 2704 |
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