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| Mirrors > Home > MPE Home > Th. List > moeu | Structured version Visualization version GIF version | ||
| Description: Uniqueness is equivalent to existence implying unique existence. Alternate definition of the at-most-one quantifier, in terms of the existential quantifier and the unique existential quantifier. (Contributed by NM, 8-Mar-1995.) This used to be the definition of the at-most-one quantifier, while df-mo 2540 was then proved as dfmo 2596. (Revised by BJ, 30-Sep-2022.) | 
| Ref | Expression | 
|---|---|
| moeu | ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | moabs 2543 | . 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑)) | |
| 2 | exmoeub 2580 | . . 3 ⊢ (∃𝑥𝜑 → (∃*𝑥𝜑 ↔ ∃!𝑥𝜑)) | |
| 3 | 2 | pm5.74i 271 | . 2 ⊢ ((∃𝑥𝜑 → ∃*𝑥𝜑) ↔ (∃𝑥𝜑 → ∃!𝑥𝜑)) | 
| 4 | 1, 3 | bitri 275 | 1 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∃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 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-mo 2540 df-eu 2569 | 
| This theorem is referenced by: dfeu 2595 dfmo 2596 sb8mo 2601 2euexv 2631 2euex 2641 2eu1 2651 2eu1v 2652 rmo5 3400 funeu 6591 dffun8 6594 modom 9280 climmo 15593 rmoxfrd 32512 nmotru 36409 bj-moeub 36850 wl-sb8mot 37581 wl-sb8motv 37582 nexmo1 38249 moeu2 38363 moxfr 42703 funressneu 47059 funressndmafv2rn 47235 | 
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