Theorem rmo5 2729

Description: Restricted "at most one" in term of uniqueness. (Contributed by NM, 16-Jun-2017.)

Assertion

Ref	Expression
rmo5	|- ( E* x e. A ph <-> ( E. x e. A ph -> E! x e. A ph ) )

Proof of Theorem rmo5

Step	Hyp	Ref	Expression
1		df-mo 2059	. 2 |- ( E* x ( x e. A /\ ph ) <-> ( E. x ( x e. A /\ ph ) -> E! x ( x e. A /\ ph ) ) )
2		df-rmo 2494	. 2 |- ( E* x e. A ph <-> E* x ( x e. A /\ ph ) )
3		df-rex 2492	. . 3 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
4		df-reu 2493	. . 3 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
5	3, 4	imbi12i 239	. 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:   -> wi 4   /\ wa 104   <-> wb 105   E.wex 1516   E!weu 2055   E*wmo 2056   e. wcel 2178   E.wrex 2487   E!wreu 2488   E*wrmo 2489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117  df-mo 2059  df-rex 2492  df-reu 2493  df-rmo 2494

This theorem is referenced by:  nrexrmo  2730  cbvrmo  2741  bdrmo  15991