Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  exmoeu Structured version   Visualization version   GIF version

Theorem exmoeu 2641
 Description: Existence is equivalent to uniqueness implying existential uniqueness. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Wolf Lammen, 5-Dec-2018.) (Proof shortened by BJ, 7-Oct-2022.)
Assertion
Ref Expression
exmoeu (∃𝑥𝜑 ↔ (∃*𝑥𝜑 → ∃!𝑥𝜑))

Proof of Theorem exmoeu
StepHypRef Expression
1 exmoeub 2640 . . 3 (∃𝑥𝜑 → (∃*𝑥𝜑 ↔ ∃!𝑥𝜑))
21biimpd 232 . 2 (∃𝑥𝜑 → (∃*𝑥𝜑 → ∃!𝑥𝜑))
3 nexmo 2599 . . . 4 (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)
43con1i 149 . . 3 (¬ ∃*𝑥𝜑 → ∃𝑥𝜑)
5 euex 2637 . . 3 (∃!𝑥𝜑 → ∃𝑥𝜑)
64, 5ja 189 . 2 ((∃*𝑥𝜑 → ∃!𝑥𝜑) → ∃𝑥𝜑)
72, 6impbii 212 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: (None)
 Copyright terms: Public domain W3C validator