Theorem eu5 2130

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 2112	. . 3 |- ( E! x ph -> E. x ph )
2		eumo 2114	. . 3 |- ( E! x ph -> E* x ph )
3	1, 2	jca 306	. 2 |- ( E! x ph -> ( E. x ph /\ E* x ph ) )
4		df-mo 2086	. . . . 5 |- ( E* x ph <-> ( E. x ph -> E! x ph ) )
5	4	biimpi 120	. . . 4 |- ( E* x ph -> ( E. x ph -> E! x ph ) )
6	5	imp 124	. . 3 |- ( ( E* x ph /\ E. x ph ) -> E! x ph )
7	6	ancoms 268	. 2 |- ( ( E. x ph /\ E* x ph ) -> E! x ph )
8	3, 7	impbii 126	1 |- ( E! x ph <-> ( E. x ph /\ E* x ph ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   E.wex 1541   E!weu 2082   E*wmo 2083

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086

This theorem is referenced by:  exmoeu2  2131  euan  2139  eu4  2145  euim  2151  euexex  2168  2euex  2170  2euswapdc  2174  2exeu  2175  reu5  2764  reuss2  3505  funcnv3  5423  fnres  5480  fnopabg  5487  brprcneu  5668  dff3im  5827  recmulnqg  7722  uptx  15265