Theorem reu5 2751

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 2127	. 2 |- ( E! x ( x e. A /\ ph ) <-> ( E. x ( x e. A /\ ph ) /\ E* x ( x e. A /\ ph ) ) )
2		df-reu 2517	. 2 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
3		df-rex 2516	. . 3 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
4		df-rmo 2518	. . 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 1540   E!weu 2079   E*wmo 2080   e. wcel 2202   E.wrex 2511   E!wreu 2512   E*wrmo 2513

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-rex 2516  df-reu 2517  df-rmo 2518

This theorem is referenced by:  reurex  2752  reurmo  2753  cbvreuw  2762  reu4  3000  reueq  3005  reusv1  4555  fncnv  5396  moriotass  6002  supeuti  7193  infeuti  7228  lteupri  7837  elrealeu  8049  rereceu  8109  exbtwnz  10511  rersqreu  11606  divalglemeunn  12500  divalglemeuneg  12502  bezoutlemeu  12596  pw2dvdseu  12758  ismgmid  13478  mndideu  13527  dedekindeu  15366  dedekindicclemicc  15375