Theorem rmo5 3162

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 2475	. 2 |- ( E* x ( x e. A /\ ph ) <-> ( E. x ( x e. A /\ ph ) -> E! x ( x e. A /\ ph ) ) )
2		df-rmo 2920	. 2 |- ( E* x e. A ph <-> E* x ( x e. A /\ ph ) )
3		df-rex 2918	. . 3 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
4		df-reu 2919	. . 3 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
5	3, 4	imbi12i 340	. 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 292	1 |- ( E* x e. A ph <-> ( E. x e. A ph -> E! x e. A ph ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   E.wex 1704   e. wcel 1990   E!weu 2470   E*wmo 2471   E.wrex 2913   E!wreu 2914   E*wrmo 2915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197  df-mo 2475  df-rex 2918  df-reu 2919  df-rmo 2920

This theorem is referenced by:  nrexrmo  3163  cbvrmo  3170  ddemeas  30299  2reurex  41181  iccpartdisj  41373