Theorem eu5 2061

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 2044	. . 3 |- ( E! x ph -> E. x ph )
2		eumo 2046	. . 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 2018	. . . . 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 1480   E!weu 2014   E*wmo 2015

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018

This theorem is referenced by:  exmoeu2  2062  euan  2070  eu4  2076  euim  2082  euexex  2099  2euex  2101  2euswapdc  2105  2exeu  2106  reu5  2678  reuss2  3402  funcnv3  5250  fnres  5304  fnopabg  5311  brprcneu  5479  dff3im  5630  recmulnqg  7332  uptx  12914