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Mirrors > Home > MPE Home > Th. List > 2exeuv | Structured version Visualization version GIF version |
Description: Double existential uniqueness implies double unique existential quantification. Version of 2exeu 2636 with 𝑥 and 𝑦 distinct, but not requiring ax-13 2365. (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 2566 | . . . 4 ⊢ (∃!𝑥∃𝑦𝜑 → ∃*𝑥∃𝑦𝜑) | |
2 | euex 2565 | . . . . 5 ⊢ (∃!𝑦𝜑 → ∃𝑦𝜑) | |
3 | 2 | moimi 2533 | . . . 4 ⊢ (∃*𝑥∃𝑦𝜑 → ∃*𝑥∃!𝑦𝜑) |
4 | 1, 3 | syl 17 | . . 3 ⊢ (∃!𝑥∃𝑦𝜑 → ∃*𝑥∃!𝑦𝜑) |
5 | 2euexv 2621 | . . 3 ⊢ (∃!𝑦∃𝑥𝜑 → ∃𝑥∃!𝑦𝜑) | |
6 | 4, 5 | anim12ci 613 | . 2 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) → (∃𝑥∃!𝑦𝜑 ∧ ∃*𝑥∃!𝑦𝜑)) |
7 | df-eu 2557 | . 2 ⊢ (∃!𝑥∃!𝑦𝜑 ↔ (∃𝑥∃!𝑦𝜑 ∧ ∃*𝑥∃!𝑦𝜑)) | |
8 | 6, 7 | sylibr 233 | 1 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) → ∃!𝑥∃!𝑦𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∃wex 1773 ∃*wmo 2526 ∃!weu 2556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-10 2129 ax-11 2146 ax-12 2163 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-nf 1778 df-mo 2528 df-eu 2557 |
This theorem is referenced by: 2eu1v 2641 |
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