Theorem moaneu 2625

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

Step	Hyp	Ref	Expression
1		moanmo 2624	. 2 ⊢ ∃*𝑥(𝜑 ∧ ∃*𝑥𝜑)
2		eumo 2578	. . . 4 ⊢ (∃!𝑥𝜑 → ∃*𝑥𝜑)
3	2	anim2i 617	. . 3 ⊢ ((𝜑 ∧ ∃!𝑥𝜑) → (𝜑 ∧ ∃*𝑥𝜑))
4	3	moimi 2545	. 2 ⊢ (∃*𝑥(𝜑 ∧ ∃*𝑥𝜑) → ∃*𝑥(𝜑 ∧ ∃!𝑥𝜑))
5	1, 4	ax-mp 5	1 ⊢ ∃*𝑥(𝜑 ∧ ∃!𝑥𝜑)

Syntax hints:   ∧ wa 396  ∃*wmo 2538  ∃!weu 2568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-11 2154  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ex 1783  df-nf 1787  df-mo 2540  df-eu 2569

This theorem is referenced by: (None)