Theorem reu5 2682

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 2066	. 2 |- ( E! x ( x e. A /\ ph ) <-> ( E. x ( x e. A /\ ph ) /\ E* x ( x e. A /\ ph ) ) )
2		df-reu 2455	. 2 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
3		df-rex 2454	. . 3 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
4		df-rmo 2456	. . 3 |- ( E* x e. A ph <-> E* x ( x e. A /\ ph ) )
5	3, 4	anbi12i 457	. 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 1485   E!weu 2019   E*wmo 2020   e. wcel 2141   E.wrex 2449   E!wreu 2450   E*wrmo 2451

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-rex 2454  df-reu 2455  df-rmo 2456

This theorem is referenced by:  reurex  2683  reurmo  2684  reu4  2924  reueq  2929  reusv1  4441  fncnv  5262  moriotass  5835  supeuti  6969  infeuti  7004  lteupri  7572  elrealeu  7784  rereceu  7844  exbtwnz  10200  rersqreu  10985  divalglemeunn  11873  divalglemeuneg  11875  bezoutlemeu  11955  pw2dvdseu  12115  ismgmid  12624  mndideu  12655  dedekindeu  13360  dedekindicclemicc  13369