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Theorem moaneu 2697
Description: Nested at-most-one and unique existential quantifiers. (Contributed by NM, 25-Jan-2006.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)
Assertion
Ref Expression
moaneu ∃*𝑥(𝜑 ∧ ∃!𝑥𝜑)

Proof of Theorem moaneu
StepHypRef Expression
1 moanmo 2696 . 2 ∃*𝑥(𝜑 ∧ ∃*𝑥𝜑)
2 eumo 2661 . . . 4 (∃!𝑥𝜑 → ∃*𝑥𝜑)
32anim2i 605 . . 3 ((𝜑 ∧ ∃!𝑥𝜑) → (𝜑 ∧ ∃*𝑥𝜑))
43moimi 2683 . 2 (∃*𝑥(𝜑 ∧ ∃*𝑥𝜑) → ∃*𝑥(𝜑 ∧ ∃!𝑥𝜑))
51, 4ax-mp 5 1 ∃*𝑥(𝜑 ∧ ∃!𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wa 384  ∃!weu 2632  ∃*wmo 2633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-10 2187  ax-11 2203  ax-12 2216
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1865  df-eu 2636  df-mo 2637
This theorem is referenced by: (None)
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