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Theorem bj-moeub 34058
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 2663 . 2 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
2 euex 2657 . . . 4 (∃!𝑥𝜑 → ∃𝑥𝜑)
3 impbi 209 . . . 4 ((∃𝑥𝜑 → ∃!𝑥𝜑) → ((∃!𝑥𝜑 → ∃𝑥𝜑) → (∃𝑥𝜑 ↔ ∃!𝑥𝜑)))
42, 3mpi 20 . . 3 ((∃𝑥𝜑 → ∃!𝑥𝜑) → (∃𝑥𝜑 ↔ ∃!𝑥𝜑))
5 biimp 216 . . 3 ((∃𝑥𝜑 ↔ ∃!𝑥𝜑) → (∃𝑥𝜑 → ∃!𝑥𝜑))
64, 5impbii 210 . 2 ((∃𝑥𝜑 → ∃!𝑥𝜑) ↔ (∃𝑥𝜑 ↔ ∃!𝑥𝜑))
71, 6bitri 276 1 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 ↔ ∃!𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wex 1773  ∃*wmo 2613  ∃!weu 2648
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774  df-mo 2615  df-eu 2649
This theorem is referenced by: (None)
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