# Metamath Proof Explorer

## Theorem rmo5

Description: Restricted "at most one" in term of uniqueness. (Contributed by NM, 16-Jun-2017)

Ref Expression
Assertion rmo5 ${⊢}{\exists }^{*}{x}\in {A}{\phi }↔\left(\exists {x}\in {A}\phantom{\rule{.4em}{0ex}}{\phi }\to \exists !{x}\in {A}\phantom{\rule{.4em}{0ex}}{\phi }\right)$

### Proof

Step Hyp Ref Expression
1 moeu ${⊢}{\exists }^{*}{x}\phantom{\rule{.4em}{0ex}}\left({x}\in {A}\wedge {\phi }\right)↔\left(\exists {x}\phantom{\rule{.4em}{0ex}}\left({x}\in {A}\wedge {\phi }\right)\to \exists !{x}\phantom{\rule{.4em}{0ex}}\left({x}\in {A}\wedge {\phi }\right)\right)$
2 df-rmo ${⊢}{\exists }^{*}{x}\in {A}{\phi }↔{\exists }^{*}{x}\phantom{\rule{.4em}{0ex}}\left({x}\in {A}\wedge {\phi }\right)$
3 df-rex ${⊢}\exists {x}\in {A}\phantom{\rule{.4em}{0ex}}{\phi }↔\exists {x}\phantom{\rule{.4em}{0ex}}\left({x}\in {A}\wedge {\phi }\right)$
4 df-reu ${⊢}\exists !{x}\in {A}\phantom{\rule{.4em}{0ex}}{\phi }↔\exists !{x}\phantom{\rule{.4em}{0ex}}\left({x}\in {A}\wedge {\phi }\right)$
5 3 4 imbi12i ${⊢}\left(\exists {x}\in {A}\phantom{\rule{.4em}{0ex}}{\phi }\to \exists !{x}\in {A}\phantom{\rule{.4em}{0ex}}{\phi }\right)↔\left(\exists {x}\phantom{\rule{.4em}{0ex}}\left({x}\in {A}\wedge {\phi }\right)\to \exists !{x}\phantom{\rule{.4em}{0ex}}\left({x}\in {A}\wedge {\phi }\right)\right)$
6 1 2 5 3bitr4i ${⊢}{\exists }^{*}{x}\in {A}{\phi }↔\left(\exists {x}\in {A}\phantom{\rule{.4em}{0ex}}{\phi }\to \exists !{x}\in {A}\phantom{\rule{.4em}{0ex}}{\phi }\right)$