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Theorem moaneu 1976
 Description: Nested "at most one" and uniqueness quantifiers. (Contributed by NM, 25-Jan-2006.)
Assertion
Ref Expression
moaneu ∃*𝑥(𝜑 ∧ ∃!𝑥𝜑)

Proof of Theorem moaneu
StepHypRef Expression
1 eumo 1932 . . 3 (∃!𝑥𝜑 → ∃*𝑥𝜑)
2 nfeu1 1911 . . . 4 𝑥∃!𝑥𝜑
32moanim 1974 . . 3 (∃*𝑥(∃!𝑥𝜑𝜑) ↔ (∃!𝑥𝜑 → ∃*𝑥𝜑))
41, 3mpbir 134 . 2 ∃*𝑥(∃!𝑥𝜑𝜑)
5 ancom 253 . . 3 ((𝜑 ∧ ∃!𝑥𝜑) ↔ (∃!𝑥𝜑𝜑))
65mobii 1937 . 2 (∃*𝑥(𝜑 ∧ ∃!𝑥𝜑) ↔ ∃*𝑥(∃!𝑥𝜑𝜑))
74, 6mpbir 134 1 ∃*𝑥(𝜑 ∧ ∃!𝑥𝜑)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∃!weu 1900  ∃*wmo 1901 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904 This theorem is referenced by: (None)
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