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Theorem moaneu 2082
 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 2038 . . 3 (∃!𝑥𝜑 → ∃*𝑥𝜑)
2 nfeu1 2017 . . . 4 𝑥∃!𝑥𝜑
32moanim 2080 . . 3 (∃*𝑥(∃!𝑥𝜑𝜑) ↔ (∃!𝑥𝜑 → ∃*𝑥𝜑))
41, 3mpbir 145 . 2 ∃*𝑥(∃!𝑥𝜑𝜑)
5 ancom 264 . . 3 ((𝜑 ∧ ∃!𝑥𝜑) ↔ (∃!𝑥𝜑𝜑))
65mobii 2043 . 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 2006  ∃*wmo 2007 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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515 This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010 This theorem is referenced by: (None)
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