Theorem eu5 1947

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 1930	. . 3 |- ( E! x ph -> E. x ph )
2		eumo 1932	. . 3 |- ( E! x ph -> E* x ph )
3	1, 2	jca 290	. 2 |- ( E! x ph -> ( E. x ph /\ E* x ph ) )
4		df-mo 1904	. . . . 5 |- ( E* x ph <-> ( E. x ph -> E! x ph ) )
5	4	biimpi 113	. . . 4 |- ( E* x ph -> ( E. x ph -> E! x ph ) )
6	5	imp 115	. . 3 |- ( ( E* x ph /\ E. x ph ) -> E! x ph )
7	6	ancoms 255	. 2 |- ( ( E. x ph /\ E* x ph ) -> E! x ph )
8	3, 7	impbii 117	1 |- ( E! x ph <-> ( E. x ph /\ E* x ph ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   E.wex 1381   E!weu 1900   E*wmo 1901

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904

This theorem is referenced by:  exmoeu2  1948  euan  1956  eu4  1962  euim  1968  euexex  1985  2euex  1987  2euswapdc  1991  2exeu  1992  reu5  2522  reuss2  3217  funcnv3  4961  fnres  5015  fnopabg  5022  brprcneu  5171  dff3im  5312  recmulnqg  6489