Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-moeub Structured version   Visualization version   GIF version

Theorem bj-moeub 36764
Description: Uniqueness is equivalent to existence being equivalent to unique existence. (Contributed by BJ, 14-Oct-2022.)
Assertion
Ref Expression
bj-moeub (∃*𝑥𝜑 ↔ (∃𝑥𝜑 ↔ ∃!𝑥𝜑))

Proof of Theorem bj-moeub
StepHypRef Expression
1 moeu 2580 . 2 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
2 euex 2574 . . . 4 (∃!𝑥𝜑 → ∃𝑥𝜑)
3 impbi 208 . . . 4 ((∃𝑥𝜑 → ∃!𝑥𝜑) → ((∃!𝑥𝜑 → ∃𝑥𝜑) → (∃𝑥𝜑 ↔ ∃!𝑥𝜑)))
42, 3mpi 20 . . 3 ((∃𝑥𝜑 → ∃!𝑥𝜑) → (∃𝑥𝜑 ↔ ∃!𝑥𝜑))
5 biimp 215 . . 3 ((∃𝑥𝜑 ↔ ∃!𝑥𝜑) → (∃𝑥𝜑 → ∃!𝑥𝜑))
64, 5impbii 209 . 2 ((∃𝑥𝜑 → ∃!𝑥𝜑) ↔ (∃𝑥𝜑 ↔ ∃!𝑥𝜑))
71, 6bitri 275 1 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 ↔ ∃!𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wex 1777  ∃*wmo 2535  ∃!weu 2565
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 2537  df-eu 2566
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator