Theorem moaneu 2102

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 2058	. . 3 ⊢ (∃!𝑥𝜑 → ∃*𝑥𝜑)
2		nfeu1 2037	. . . 4 ⊢ Ⅎ𝑥∃!𝑥𝜑
3	2	moanim 2100	. . 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 2063	. 2 ⊢ (∃*𝑥(𝜑 ∧ ∃!𝑥𝜑) ↔ ∃*𝑥(∃!𝑥𝜑 ∧ 𝜑))
7	4, 6	mpbir 146	1 ⊢ ∃*𝑥(𝜑 ∧ ∃!𝑥𝜑)

Syntax hints:   → wi 4   ∧ wa 104  ∃!weu 2026  ∃*wmo 2027

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030

This theorem is referenced by: (None)