Theorem reu5 2711

Description: Restricted uniqueness in terms of "at most one". (Contributed by NM, 23-May-1999.) (Revised by NM, 16-Jun-2017.)

Assertion

Ref	Expression
reu5	|- ( E! x e. A ph <-> ( E. x e. A ph /\ E* x e. A ph ) )

Proof of Theorem reu5

Step	Hyp	Ref	Expression
1		eu5 2089	. 2 |- ( E! x ( x e. A /\ ph ) <-> ( E. x ( x e. A /\ ph ) /\ E* x ( x e. A /\ ph ) ) )
2		df-reu 2479	. 2 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
3		df-rex 2478	. . 3 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
4		df-rmo 2480	. . 3 |- ( E* x e. A ph <-> E* x ( x e. A /\ ph ) )
5	3, 4	anbi12i 460	. 2 |- ( ( E. x e. A ph /\ E* x e. A ph ) <-> ( E. x ( x e. A /\ ph ) /\ E* x ( x e. A /\ ph ) ) )
6	1, 2, 5	3bitr4i 212	1 |- ( E! x e. A ph <-> ( E. x e. A ph /\ E* x e. A ph ) )

Syntax hints:   /\ wa 104   <-> wb 105   E.wex 1503   E!weu 2042   E*wmo 2043   e. wcel 2164   E.wrex 2473   E!wreu 2474   E*wrmo 2475

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-rex 2478  df-reu 2479  df-rmo 2480

This theorem is referenced by:  reurex  2712  reurmo  2713  reu4  2955  reueq  2960  reusv1  4490  fncnv  5321  moriotass  5903  supeuti  7055  infeuti  7090  lteupri  7679  elrealeu  7891  rereceu  7951  exbtwnz  10322  rersqreu  11175  divalglemeunn  12065  divalglemeuneg  12067  bezoutlemeu  12147  pw2dvdseu  12309  ismgmid  12963  mndideu  13010  dedekindeu  14802  dedekindicclemicc  14811