Theorem moanmo 2625

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 22	. . 3 ⊢ (∃*𝑥𝜑 → ∃*𝑥𝜑)
2		nfmo1 2560	. . . 4 ⊢ Ⅎ𝑥∃*𝑥𝜑
3	2	moanim 2623	. . 3 ⊢ (∃*𝑥(∃*𝑥𝜑 ∧ 𝜑) ↔ (∃*𝑥𝜑 → ∃*𝑥𝜑))
4	1, 3	mpbir 231	. 2 ⊢ ∃*𝑥(∃*𝑥𝜑 ∧ 𝜑)
5		ancom 460	. . 3 ⊢ ((𝜑 ∧ ∃*𝑥𝜑) ↔ (∃*𝑥𝜑 ∧ 𝜑))
6	5	mobii 2551	. 2 ⊢ (∃*𝑥(𝜑 ∧ ∃*𝑥𝜑) ↔ ∃*𝑥(∃*𝑥𝜑 ∧ 𝜑))
7	4, 6	mpbir 231	1 ⊢ ∃*𝑥(𝜑 ∧ ∃*𝑥𝜑)

Syntax hints:   → wi 4   ∧ wa 395  ∃*wmo 2541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-10 2141  ax-11 2158  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-ex 1778  df-nf 1782  df-mo 2543

This theorem is referenced by:  moaneu  2626