Theorem exmoeu 2246

Description: Existence in terms of "at most one" and uniqueness. (Contributed by NM, 5-Apr-2004.)

Assertion

Ref	Expression
exmoeu	⊢ (∃xφ ↔ (∃*xφ → ∃!xφ))

Proof of Theorem exmoeu

Step	Hyp	Ref	Expression
1		df-mo 2209	. . . 4 ⊢ (∃*xφ ↔ (∃xφ → ∃!xφ))
2	1	biimpi 186	. . 3 ⊢ (∃*xφ → (∃xφ → ∃!xφ))
3	2	com12 27	. 2 ⊢ (∃xφ → (∃*xφ → ∃!xφ))
4	1	biimpri 197	. . . 4 ⊢ ((∃xφ → ∃!xφ) → ∃*xφ)
5		euex 2227	. . . 4 ⊢ (∃!xφ → ∃xφ)
6	4, 5	imim12i 53	. . 3 ⊢ ((∃*xφ → ∃!xφ) → ((∃xφ → ∃!xφ) → ∃xφ))
7		peirce 172	. . 3 ⊢ (((∃xφ → ∃!xφ) → ∃xφ) → ∃xφ)
8	6, 7	syl 15	. 2 ⊢ ((∃*xφ → ∃!xφ) → ∃xφ)
9	3, 8	impbii 180	1 ⊢ (∃xφ ↔ (∃*xφ → ∃!xφ))

Syntax hints:   → wi 4   ↔ wb 176  ∃wex 1541  ∃!weu 2204  ∃*wmo 2205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209

This theorem is referenced by: (None)