Theorem moanmo 2652

Description: Nested at-most-one quantifiers. (Contributed by NM, 25-Jan-2006.)

Assertion

Ref	Expression
moanmo	⊢ ∃*𝑥(𝜑 ∧ ∃*𝑥𝜑)

Proof of Theorem moanmo

Step	Hyp	Ref	Expression
1		id 23	. . 3 ⊢ (∃*𝑥𝜑 → ∃*𝑥𝜑)
2		nfmo1 2587	. . . 4 ⊢ Ⅎ𝑥∃*𝑥𝜑
3	2	moanim 2650	. . 3 ⊢ (∃*𝑥(∃*𝑥𝜑 ∧ 𝜑) ↔ (∃*𝑥𝜑 → ∃*𝑥𝜑))
4	1, 3	mpbir 234	. 2 ⊢ ∃*𝑥(∃*𝑥𝜑 ∧ 𝜑)
5		ancom 466	. . 3 ⊢ ((𝜑 ∧ ∃*𝑥𝜑) ↔ (∃*𝑥𝜑 ∧ 𝜑))
6	5	mobii 2578	. 2 ⊢ (∃*𝑥(𝜑 ∧ ∃*𝑥𝜑) ↔ ∃*𝑥(∃*𝑥𝜑 ∧ 𝜑))
7	4, 6	mpbir 234	1 ⊢ ∃*𝑥(𝜑 ∧ ∃*𝑥𝜑)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∃*wmo 2567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-11 2195  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817  df-mo 2569

This theorem is used by:  moaneu  2653