Theorem rmo5 2569

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 1945	. 2 |- ( E* x ( x e. A /\ ph ) <-> ( E. x ( x e. A /\ ph ) -> E! x ( x e. A /\ ph ) ) )
2		df-rmo 2356	. 2 |- ( E* x e. A ph <-> E* x ( x e. A /\ ph ) )
3		df-rex 2354	. . 3 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
4		df-reu 2355	. . 3 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
5	3, 4	imbi12i 237	. 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 210	1 |- ( E* x e. A ph <-> ( E. x e. A ph -> E! x e. A ph ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   E.wex 1421   e. wcel 1433   E!weu 1941   E*wmo 1942   E.wrex 2349   E!wreu 2350   E*wrmo 2351

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106

This theorem depends on definitions:  df-bi 115  df-mo 1945  df-rex 2354  df-reu 2355  df-rmo 2356

This theorem is referenced by:  nrexrmo  2570  cbvrmo  2576  bdrmo  10647