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Theorem moeu 2581
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.)
Assertion
Ref Expression
moeu (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))

Proof of Theorem moeu
StepHypRef Expression
1 moabs 2541 . 2 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑))
2 exmoeub 2578 . . 3 (∃𝑥𝜑 → (∃*𝑥𝜑 ↔ ∃!𝑥𝜑))
32pm5.74i 271 . 2 ((∃𝑥𝜑 → ∃*𝑥𝜑) ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
41, 3bitri 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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