# Metamath Proof Explorer

## Theorem reu5

Description: Restricted uniqueness in terms of "at most one." (Contributed by NM, 23-May-1999) (Revised by NM, 16-Jun-2017)

Ref Expression
Assertion reu5 ${⊢}\exists !{x}\in {A}\phantom{\rule{.4em}{0ex}}{\phi }↔\left(\exists {x}\in {A}\phantom{\rule{.4em}{0ex}}{\phi }\wedge {\exists }^{*}{x}\in {A}{\phi }\right)$

### Proof

Step Hyp Ref Expression
1 df-eu ${⊢}\exists !{x}\phantom{\rule{.4em}{0ex}}\left({x}\in {A}\wedge {\phi }\right)↔\left(\exists {x}\phantom{\rule{.4em}{0ex}}\left({x}\in {A}\wedge {\phi }\right)\wedge {\exists }^{*}{x}\phantom{\rule{.4em}{0ex}}\left({x}\in {A}\wedge {\phi }\right)\right)$
2 df-reu ${⊢}\exists !{x}\in {A}\phantom{\rule{.4em}{0ex}}{\phi }↔\exists !{x}\phantom{\rule{.4em}{0ex}}\left({x}\in {A}\wedge {\phi }\right)$
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-rmo ${⊢}{\exists }^{*}{x}\in {A}{\phi }↔{\exists }^{*}{x}\phantom{\rule{.4em}{0ex}}\left({x}\in {A}\wedge {\phi }\right)$
5 3 4 anbi12i ${⊢}\left(\exists {x}\in {A}\phantom{\rule{.4em}{0ex}}{\phi }\wedge {\exists }^{*}{x}\in {A}{\phi }\right)↔\left(\exists {x}\phantom{\rule{.4em}{0ex}}\left({x}\in {A}\wedge {\phi }\right)\wedge {\exists }^{*}{x}\phantom{\rule{.4em}{0ex}}\left({x}\in {A}\wedge {\phi }\right)\right)$
6 1 2 5 3bitr4i ${⊢}\exists !{x}\in {A}\phantom{\rule{.4em}{0ex}}{\phi }↔\left(\exists {x}\in {A}\phantom{\rule{.4em}{0ex}}{\phi }\wedge {\exists }^{*}{x}\in {A}{\phi }\right)$