Theorem reu5 2643

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 2046	. 2 |- ( E! x ( x e. A /\ ph ) <-> ( E. x ( x e. A /\ ph ) /\ E* x ( x e. A /\ ph ) ) )
2		df-reu 2423	. 2 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
3		df-rex 2422	. . 3 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
4		df-rmo 2424	. . 3 |- ( E* x e. A ph <-> E* x ( x e. A /\ ph ) )
5	3, 4	anbi12i 455	. 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 211	1 |- ( E! x e. A ph <-> ( E. x e. A ph /\ E* x e. A ph ) )

Syntax hints:   /\ wa 103   <-> wb 104   E.wex 1468   e. wcel 1480   E!weu 1999   E*wmo 2000   E.wrex 2417   E!wreu 2418   E*wrmo 2419

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-rex 2422  df-reu 2423  df-rmo 2424

This theorem is referenced by:  reurex  2644  reurmo  2645  reu4  2878  reueq  2883  reusv1  4379  fncnv  5189  moriotass  5758  supeuti  6881  infeuti  6916  lteupri  7432  elrealeu  7644  rereceu  7704  exbtwnz  10035  rersqreu  10807  divalglemeunn  11625  divalglemeuneg  11627  bezoutlemeu  11702  pw2dvdseu  11853  dedekindeu  12780  dedekindicclemicc  12789