Theorem eu5 2053

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 2036	. . 3 |- ( E! x ph -> E. x ph )
2		eumo 2038	. . 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 2010	. . . . 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 1472   E!weu 2006   E*wmo 2007

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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010

This theorem is referenced by:  exmoeu2  2054  euan  2062  eu4  2068  euim  2074  euexex  2091  2euex  2093  2euswapdc  2097  2exeu  2098  reu5  2669  reuss2  3387  funcnv3  5233  fnres  5287  fnopabg  5294  brprcneu  5462  dff3im  5613  recmulnqg  7312  uptx  12716