Theorem eu5 2044

Description: Uniqueness in terms of "at most one." (Contributed by NM, 23-Mar-1995.) (Proof rewritten by Jim Kingdon, 27-May-2018.)

Assertion

Ref	Expression
eu5	|- ( E! x ph <-> ( E. x ph /\ E* x ph ) )

Proof of Theorem eu5

Step	Hyp	Ref	Expression
1		euex 2027	. . 3 |- ( E! x ph -> E. x ph )
2		eumo 2029	. . 3 |- ( E! x ph -> E* x ph )
3	1, 2	jca 304	. 2 |- ( E! x ph -> ( E. x ph /\ E* x ph ) )
4		df-mo 2001	. . . . 5 |- ( E* x ph <-> ( E. x ph -> E! x ph ) )
5	4	biimpi 119	. . . 4 |- ( E* x ph -> ( E. x ph -> E! x ph ) )
6	5	imp 123	. . 3 |- ( ( E* x ph /\ E. x ph ) -> E! x ph )
7	6	ancoms 266	. 2 |- ( ( E. x ph /\ E* x ph ) -> E! x ph )
8	3, 7	impbii 125	1 |- ( E! x ph <-> ( E. x ph /\ E* x ph ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   E.wex 1468   E!weu 1997   E*wmo 1998

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001

This theorem is referenced by:  exmoeu2  2045  euan  2053  eu4  2059  euim  2065  euexex  2082  2euex  2084  2euswapdc  2088  2exeu  2089  reu5  2641  reuss2  3351  funcnv3  5180  fnres  5234  fnopabg  5241  brprcneu  5407  dff3im  5558  recmulnqg  7192  uptx  12432