MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  moaneu Structured version   Visualization version   GIF version

Theorem moaneu 2520
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 2519 . 2 ∃*𝑥(𝜑 ∧ ∃*𝑥𝜑)
2 eumo 2486 . . . 4 (∃!𝑥𝜑 → ∃*𝑥𝜑)
32anim2i 590 . . 3 ((𝜑 ∧ ∃!𝑥𝜑) → (𝜑 ∧ ∃*𝑥𝜑))
43moimi 2507 . 2 (∃*𝑥(𝜑 ∧ ∃*𝑥𝜑) → ∃*𝑥(𝜑 ∧ ∃!𝑥𝜑))
51, 4ax-mp 5 1 ∃*𝑥(𝜑 ∧ ∃!𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wa 382  ∃!weu 2457  ∃*wmo 2458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-eu 2461  df-mo 2462
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator