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Theorem moaneu 2702
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 2701 . 2 ∃*𝑥(𝜑 ∧ ∃*𝑥𝜑)
2 eumo 2657 . . . 4 (∃!𝑥𝜑 → ∃*𝑥𝜑)
32anim2i 618 . . 3 ((𝜑 ∧ ∃!𝑥𝜑) → (𝜑 ∧ ∃*𝑥𝜑))
43moimi 2621 . 2 (∃*𝑥(𝜑 ∧ ∃*𝑥𝜑) → ∃*𝑥(𝜑 ∧ ∃!𝑥𝜑))
51, 4ax-mp 5 1 ∃*𝑥(𝜑 ∧ ∃!𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wa 398  ∃*wmo 2614  ∃!weu 2647
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-10 2138  ax-11 2153  ax-12 2169
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1774  df-nf 1778  df-mo 2616  df-eu 2648
This theorem is referenced by: (None)
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