Theorem rmo5 2828

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 2209	. 2 |- (E*x(x e. A /\ ph) <-> (E.x(x e. A /\ ph) -> E!x(x e. A /\ ph)))
2		df-rmo 2623	. 2 |- (E*x e. A ph <-> E*x(x e. A /\ ph))
3		df-rex 2621	. . 3 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
4		df-reu 2622	. . 3 |- (E!x e. A ph <-> E!x(x e. A /\ ph))
5	3, 4	imbi12i 316	. 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 268	1 |- (E*x e. A ph <-> (E.x e. A ph -> E!x e. A ph))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  E.wex 1541   e. wcel 1710  E!weu 2204  E*wmo 2205  E.wrex 2616  E!wreu 2617  E*wrmo 2618

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 177  df-mo 2209  df-rex 2621  df-reu 2622  df-rmo 2623

This theorem is referenced by:  nrexrmo  2829  cbvrmo  2835