Theorem reu5 2749

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 2125	. 2 |- ( E! x ( x e. A /\ ph ) <-> ( E. x ( x e. A /\ ph ) /\ E* x ( x e. A /\ ph ) ) )
2		df-reu 2515	. 2 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
3		df-rex 2514	. . 3 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
4		df-rmo 2516	. . 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 1538   E!weu 2077   E*wmo 2078   e. wcel 2200   E.wrex 2509   E!wreu 2510   E*wrmo 2511

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-rex 2514  df-reu 2515  df-rmo 2516

This theorem is referenced by:  reurex  2750  reurmo  2751  cbvreuw  2760  reu4  2997  reueq  3002  reusv1  4549  fncnv  5387  moriotass  5991  supeuti  7172  infeuti  7207  lteupri  7815  elrealeu  8027  rereceu  8087  exbtwnz  10482  rersqreu  11555  divalglemeunn  12448  divalglemeuneg  12450  bezoutlemeu  12544  pw2dvdseu  12706  ismgmid  13426  mndideu  13475  dedekindeu  15313  dedekindicclemicc  15322