Theorem moaneu 2533

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

Step	Hyp	Ref	Expression
1		moanmo 2532	. 2 ⊢ ∃*𝑥(𝜑 ∧ ∃*𝑥𝜑)
2		eumo 2499	. . . 4 ⊢ (∃!𝑥𝜑 → ∃*𝑥𝜑)
3	2	anim2i 593	. . 3 ⊢ ((𝜑 ∧ ∃!𝑥𝜑) → (𝜑 ∧ ∃*𝑥𝜑))
4	3	moimi 2520	. 2 ⊢ (∃*𝑥(𝜑 ∧ ∃*𝑥𝜑) → ∃*𝑥(𝜑 ∧ ∃!𝑥𝜑))
5	1, 4	ax-mp 5	1 ⊢ ∃*𝑥(𝜑 ∧ ∃!𝑥𝜑)

Syntax hints:   ∧ wa 384  ∃!weu 2470  ∃*wmo 2471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-eu 2474  df-mo 2475

This theorem is referenced by: (None)