Theorem exmoeu2 1989

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 1988	. 2 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
2	1	baibr 862	1 ⊢ (∃𝑥𝜑 → (∃*𝑥𝜑 ↔ ∃!𝑥𝜑))

Syntax hints:   → wi 4   ↔ wb 103  ∃wex 1421  ∃!weu 1941  ∃*wmo 1942

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945

This theorem is referenced by:  n0mmoeu  3264  fneu  5023