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Theorem moaneu 2671
Description: Nested "at most one" and uniqueness 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 2670 . 2 ∃*𝑥(𝜑 ∧ ∃*𝑥𝜑)
2 eumo 2636 . . . 4 (∃!𝑥𝜑 → ∃*𝑥𝜑)
32anim2i 594 . . 3 ((𝜑 ∧ ∃!𝑥𝜑) → (𝜑 ∧ ∃*𝑥𝜑))
43moimi 2658 . 2 (∃*𝑥(𝜑 ∧ ∃*𝑥𝜑) → ∃*𝑥(𝜑 ∧ ∃!𝑥𝜑))
51, 4ax-mp 5 1 ∃*𝑥(𝜑 ∧ ∃!𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wa 383  ∃!weu 2607  ∃*wmo 2608
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-10 2168  ax-11 2183  ax-12 2196
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-eu 2611  df-mo 2612
This theorem is referenced by: (None)
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