Metamath Proof Explorer

Theorem reu5

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

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

Proof

Step Hyp Ref Expression
1 df-eu 
 |-  ( E! x ( x e. A /\ ph ) <-> ( E. x ( x e. A /\ ph ) /\ E* x ( x e. A /\ ph ) ) )
2 df-reu 
 |-  ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
3 df-rex 
 |-  ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
4 df-rmo 
 |-  ( E* x e. A ph <-> E* x ( x e. A /\ ph ) )
5 3 4 anbi12i 
 |-  ( ( 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 
 |-  ( E! x e. A ph <-> ( E. x e. A ph /\ E* x e. A ph ) )