Theorem moaneu 2082

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

Assertion

Ref	Expression
moaneu	⊢ ∃*𝑥(𝜑 ∧ ∃!𝑥𝜑)

Proof of Theorem moaneu

Step	Hyp	Ref	Expression
1		eumo 2038	. . 3 ⊢ (∃!𝑥𝜑 → ∃*𝑥𝜑)
2		nfeu1 2017	. . . 4 ⊢ Ⅎ𝑥∃!𝑥𝜑
3	2	moanim 2080	. . 3 ⊢ (∃*𝑥(∃!𝑥𝜑 ∧ 𝜑) ↔ (∃!𝑥𝜑 → ∃*𝑥𝜑))
4	1, 3	mpbir 145	. 2 ⊢ ∃*𝑥(∃!𝑥𝜑 ∧ 𝜑)
5		ancom 264	. . 3 ⊢ ((𝜑 ∧ ∃!𝑥𝜑) ↔ (∃!𝑥𝜑 ∧ 𝜑))
6	5	mobii 2043	. 2 ⊢ (∃*𝑥(𝜑 ∧ ∃!𝑥𝜑) ↔ ∃*𝑥(∃!𝑥𝜑 ∧ 𝜑))
7	4, 6	mpbir 145	1 ⊢ ∃*𝑥(𝜑 ∧ ∃!𝑥𝜑)

Syntax hints:   → wi 4   ∧ wa 103  ∃!weu 2006  ∃*wmo 2007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010

This theorem is referenced by: (None)