Theorem moaneu

Description: Nested at-most-one and unique existential quantifiers. (Contributed by NM, 25-Jan-2006) (Proof shortened by Wolf Lammen, 27-Dec-2018)

	Ref	Expression
Assertion	moaneu	⊢ ∃* 𝑥 ( 𝜑 ∧ ∃! 𝑥 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		moanmo	⊢ ∃* 𝑥 ( 𝜑 ∧ ∃* 𝑥 𝜑 )
2		eumo	⊢ ( ∃! 𝑥 𝜑 → ∃* 𝑥 𝜑 )
3	2	anim2i	⊢ ( ( 𝜑 ∧ ∃! 𝑥 𝜑 ) → ( 𝜑 ∧ ∃* 𝑥 𝜑 ) )
4	3	moimi	⊢ ( ∃* 𝑥 ( 𝜑 ∧ ∃* 𝑥 𝜑 ) → ∃* 𝑥 ( 𝜑 ∧ ∃! 𝑥 𝜑 ) )
5	1 4	ax-mp	⊢ ∃* 𝑥 ( 𝜑 ∧ ∃! 𝑥 𝜑 )