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Theorem moaneu 1992
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 1948 . . 3 (∃!𝑥𝜑 → ∃*𝑥𝜑)
2 nfeu1 1927 . . . 4 𝑥∃!𝑥𝜑
32moanim 1990 . . 3 (∃*𝑥(∃!𝑥𝜑𝜑) ↔ (∃!𝑥𝜑 → ∃*𝑥𝜑))
41, 3mpbir 138 . 2 ∃*𝑥(∃!𝑥𝜑𝜑)
5 ancom 257 . . 3 ((𝜑 ∧ ∃!𝑥𝜑) ↔ (∃!𝑥𝜑𝜑))
65mobii 1953 . 2 (∃*𝑥(𝜑 ∧ ∃!𝑥𝜑) ↔ ∃*𝑥(∃!𝑥𝜑𝜑))
74, 6mpbir 138 1 ∃*𝑥(𝜑 ∧ ∃!𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  ∃!weu 1916  ∃*wmo 1917
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920
This theorem is referenced by: (None)
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