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Theorem moaneu 2053
Description: Nested at-most-one and unique existential quantifiers. (Contributed by NM, 25-Jan-2006.)
Assertion
Ref Expression
moaneu ∃*𝑥(𝜑 ∧ ∃!𝑥𝜑)

Proof of Theorem moaneu
StepHypRef Expression
1 eumo 2009 . . 3 (∃!𝑥𝜑 → ∃*𝑥𝜑)
2 nfeu1 1988 . . . 4 𝑥∃!𝑥𝜑
32moanim 2051 . . 3 (∃*𝑥(∃!𝑥𝜑𝜑) ↔ (∃!𝑥𝜑 → ∃*𝑥𝜑))
41, 3mpbir 145 . 2 ∃*𝑥(∃!𝑥𝜑𝜑)
5 ancom 264 . . 3 ((𝜑 ∧ ∃!𝑥𝜑) ↔ (∃!𝑥𝜑𝜑))
65mobii 2014 . 2 (∃*𝑥(𝜑 ∧ ∃!𝑥𝜑) ↔ ∃*𝑥(∃!𝑥𝜑𝜑))
74, 6mpbir 145 1 ∃*𝑥(𝜑 ∧ ∃!𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  ∃!weu 1977  ∃*wmo 1978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981
This theorem is referenced by: (None)
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