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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 2538 was then proved as dfmo 2594. (Revised by BJ, 30-Sep-2022.) |
Ref | Expression |
---|---|
moeu | ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | moabs 2541 | . 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑)) | |
2 | exmoeub 2578 | . . 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 1776 ∃*wmo 2536 ∃!weu 2566 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-mo 2538 df-eu 2567 |
This theorem is referenced by: dfeu 2593 dfmo 2594 sb8mo 2599 2euexv 2629 2euex 2639 2eu1 2649 2eu1v 2650 rmo5 3398 funeu 6593 dffun8 6596 modom 9278 climmo 15590 rmoxfrd 32521 nmotru 36391 bj-moeub 36832 wl-sb8mot 37561 wl-sb8motv 37562 nexmo1 38229 moeu2 38344 moxfr 42680 funressneu 46997 funressndmafv2rn 47173 |
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