Theorem rmo5 2519

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 1901	. 2 |- (E*x(x e. A /\ ph) <-> (E.x(x e. A /\ ph) -> E!x(x e. A /\ ph)))
2		df-rmo 2308	. 2 |- (E*x e. A ph <-> E*x(x e. A /\ ph))
3		df-rex 2306	. . 3 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
4		df-reu 2307	. . 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 1378   e. wcel 1390  E!weu 1897  E*wmo 1898  E.wrex 2301  E!wreu 2302  E*wrmo 2303

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 1901  df-rex 2306  df-reu 2307  df-rmo 2308

This theorem is referenced by:  nrexrmo  2520  cbvrmo  2526  bdrmo  9291