Theorem exmoeu2 2247

Description: Existence implies "at most one" is equivalent to uniqueness. (Contributed by NM, 5-Apr-2004.)

Assertion

Ref	Expression
exmoeu2	⊢ (∃xφ → (∃*xφ ↔ ∃!xφ))

Proof of Theorem exmoeu2

Step	Hyp	Ref	Expression
1		eu5 2242	. 2 ⊢ (∃!xφ ↔ (∃xφ ∧ ∃*xφ))
2	1	baibr 872	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:  dffun8  5134  fneu  5187