Theorem eu5 2092

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 2075	. . 3 |- ( E! x ph -> E. x ph )
2		eumo 2077	. . 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 2049	. . . . 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 1506   E!weu 2045   E*wmo 2046

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049

This theorem is referenced by:  exmoeu2  2093  euan  2101  eu4  2107  euim  2113  euexex  2130  2euex  2132  2euswapdc  2136  2exeu  2137  reu5  2714  reuss2  3443  funcnv3  5320  fnres  5374  fnopabg  5381  brprcneu  5551  dff3im  5707  recmulnqg  7458  uptx  14510