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Theorem moaneu 2024
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 1980 . . 3 (∃!𝑥𝜑 → ∃*𝑥𝜑)
2 nfeu1 1959 . . . 4 𝑥∃!𝑥𝜑
32moanim 2022 . . 3 (∃*𝑥(∃!𝑥𝜑𝜑) ↔ (∃!𝑥𝜑 → ∃*𝑥𝜑))
41, 3mpbir 144 . 2 ∃*𝑥(∃!𝑥𝜑𝜑)
5 ancom 262 . . 3 ((𝜑 ∧ ∃!𝑥𝜑) ↔ (∃!𝑥𝜑𝜑))
65mobii 1985 . 2 (∃*𝑥(𝜑 ∧ ∃!𝑥𝜑) ↔ ∃*𝑥(∃!𝑥𝜑𝜑))
74, 6mpbir 144 1 ∃*𝑥(𝜑 ∧ ∃!𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  ∃!weu 1948  ∃*wmo 1949
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952
This theorem is referenced by: (None)
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