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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 2598 was then proved as dfmo 2657. (Revised by BJ, 30-Sep-2022.) |
Ref | Expression |
---|---|
moeu | ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | moabs 2601 | . 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑)) | |
2 | exmoeub 2640 | . . 3 ⊢ (∃𝑥𝜑 → (∃*𝑥𝜑 ↔ ∃!𝑥𝜑)) | |
3 | 2 | pm5.74i 274 | . 2 ⊢ ((∃𝑥𝜑 → ∃*𝑥𝜑) ↔ (∃𝑥𝜑 → ∃!𝑥𝜑)) |
4 | 1, 3 | bitri 278 | 1 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∃wex 1781 ∃*wmo 2596 ∃!weu 2628 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-mo 2598 df-eu 2629 |
This theorem is referenced by: dfeu 2656 dfmo 2657 sb8mo 2662 cbvmowOLD 2664 2euexv 2693 2euex 2703 2eu1 2712 2eu1v 2713 rmo5 3379 funeu 6349 dffun8 6352 modom 8703 climmo 14906 rmoxfrd 30264 nmotru 33869 amosym1 33887 bj-moeub 34288 wl-sb8mot 34979 nexmo1 35668 moxfr 39633 funressneu 43639 funressndmafv2rn 43779 |
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