Theorem eu5 2125

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 2107	. . 3 |- ( E! x ph -> E. x ph )
2		eumo 2109	. . 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 2081	. . . . 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 1538   E!weu 2077   E*wmo 2078

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081

This theorem is referenced by:  exmoeu2  2126  euan  2134  eu4  2140  euim  2146  euexex  2163  2euex  2165  2euswapdc  2169  2exeu  2170  reu5  2749  reuss2  3484  funcnv3  5383  fnres  5440  fnopabg  5447  brprcneu  5620  dff3im  5780  recmulnqg  7578  uptx  14948