Theorem moeu 2586

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 2543 was then proved as dfmo 2599. (Revised by BJ, 30-Sep-2022.)

Assertion

Ref	Expression
moeu	⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))

Proof of Theorem moeu

Step	Hyp	Ref	Expression
1		moabs 2546	. 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑))
2		exmoeub 2583	. . 3 ⊢ (∃𝑥𝜑 → (∃*𝑥𝜑 ↔ ∃!𝑥𝜑))
3	2	pm5.74i 271	. 2 ⊢ ((∃𝑥𝜑 → ∃*𝑥𝜑) ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
4	1, 3	bitri 275	1 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))

Syntax hints:   → wi 4   ↔ wb 206  ∃wex 1777  ∃*wmo 2541  ∃!weu 2571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-mo 2543  df-eu 2572

This theorem is referenced by:  dfeu  2598  dfmo  2599  sb8mo  2604  2euexv  2634  2euex  2644  2eu1  2654  2eu1v  2655  rmo5  3408  funeu  6603  dffun8  6606  modom  9307  climmo  15603  rmoxfrd  32521  nmotru  36374  bj-moeub  36815  wl-sb8mot  37534  wl-sb8motv  37535  nexmo1  38203  moeu2  38318  moxfr  42648  funressneu  46962  funressndmafv2rn  47138