Theorem moaneu 2156

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 2111	. . 3 ⊢ (∃!𝑥𝜑 → ∃*𝑥𝜑)
2		nfeu1 2090	. . . 4 ⊢ Ⅎ𝑥∃!𝑥𝜑
3	2	moanim 2154	. . 3 ⊢ (∃*𝑥(∃!𝑥𝜑 ∧ 𝜑) ↔ (∃!𝑥𝜑 → ∃*𝑥𝜑))
4	1, 3	mpbir 146	. 2 ⊢ ∃*𝑥(∃!𝑥𝜑 ∧ 𝜑)
5		ancom 266	. . 3 ⊢ ((𝜑 ∧ ∃!𝑥𝜑) ↔ (∃!𝑥𝜑 ∧ 𝜑))
6	5	mobii 2116	. 2 ⊢ (∃*𝑥(𝜑 ∧ ∃!𝑥𝜑) ↔ ∃*𝑥(∃!𝑥𝜑 ∧ 𝜑))
7	4, 6	mpbir 146	1 ⊢ ∃*𝑥(𝜑 ∧ ∃!𝑥𝜑)

Syntax hints:   → wi 4   ∧ wa 104  ∃!weu 2079  ∃*wmo 2080

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083

This theorem is referenced by: (None)