Theorem rmo5 2525

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 1904	. 2 |- ( E* x ( x e. A /\ ph ) <-> ( E. x ( x e. A /\ ph ) -> E! x ( x e. A /\ ph ) ) )
2		df-rmo 2314	. 2 |- ( E* x e. A ph <-> E* x ( x e. A /\ ph ) )
3		df-rex 2312	. . 3 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
4		df-reu 2313	. . 3 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
5	3, 4	imbi12i 228	. 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 201	1 |- ( E* x e. A ph <-> ( E. x e. A ph -> E! x e. A ph ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   E.wex 1381   e. wcel 1393   E!weu 1900   E*wmo 1901   E.wrex 2307   E!wreu 2308   E*wrmo 2309

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110  df-mo 1904  df-rex 2312  df-reu 2313  df-rmo 2314

This theorem is referenced by:  nrexrmo  2526  cbvrmo  2532  bdrmo  9976